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ac{1}{k} \sum_{j=1}^k {\mathbb{E}}\left({\left\Vert X_j \right\Vert}^2
{\mathbbm{1}_{{\left\Vert X_j \right\Vert} > {\varepsilon}\sqrt{k}}} \right).$$ Then $$d_{BL} \left(\frac{1}{\sqrt{k}} \sum_{j=1}^k X_j, Z\right) \le
C_d \inf_{0 \le {\varepsilon}\le 1} ({\varepsilon}+ \theta({\varepsilon})),$$ where $Z$ is a standard Gaussian random vector in ${\mathbb{R}}^d$, and $C_d >
0$ depends only on $d$.
Let $f:{\mathbb{C}}\to {\mathbb{R}}$ with ${\left\Vert f \right\Vert}_{BL} \le 1$. Observe that $$\label{E:mean-int}
{\mathbb{E}}\mu(f) =
\frac{1}{{\left\vert G \right\vert}} \sum_{\chi \in \widehat{G}} {\mathbb{E}}f(\lambda_\chi)
= \frac{1}{{\left\vert G \right\vert}} \sum_{\chi \in \widehat{G}}
{\mathbb{E}}f\left(\frac{1}{\sqrt{{\left\vert G \right\vert}}}\sum_{a \in G} \chi(a) Y_a\right),$$ where $(n)$ superscripts have been omitted for simplicity. We consider $\lambda_\chi$ as a sum of independent random vectors in ${\mathbb{R}}^2 \cong {\mathbb{C}}$. The relevant covariances are $$\operatorname{Cov}(\chi(a) Y_a) = \begin{bmatrix}
{\mathbb{E}}({\operatorname{Re}}\chi(a) Y_a)^2 & {\mathbb{E}}({\operatorname{Re}}\chi(a) Y_a) ({\operatorname{Im}}\chi(a) Y_a) \\
{\mathbb{E}}({\operatorname{Re}}\chi(a) Y_a) ({\operatorname{Im}}\chi(a) Y_a) & {\mathbb{E}}({\operatorname{Im}}\chi(a) Y_a)^2
\end{bmatrix}.$$ The identities $$\begin{split}\label{E:Re-Im}
({\operatorname{Re}}w)({\operatorname{Re}}z)
&= \tfrac{1}{2} {\operatorname{Re}}\bigl[(w + \overline{w}) z \bigr], \\
({\operatorname{Im}}w)({\operatorname{Im}}z)
&= \tfrac{1}{2} {\operatorname{Re}}\bigl[(\overline{w} - w) z \bigr], \\
({\operatorname{Re}}w)({\operatorname{Im}}z) &= \tfrac{1}{2} {\operatorname{Im}}\bigl[(w - \overline{w}) z
\bigr],
\end{split}$$ will be useful.
Setting $w = z = \chi(a) Y_a$ for a fixed $\chi \in \widehat{G}$, $$\begin{aligned}
\sum_{a \in G} {\mathbb{E}}({\operatorname{Re}}\chi(a) Y_a)^2 &=
\sum_{a \in G} \left[\frac{1}{2} {\operatorname{Re}}{\mathbb{E}}\le
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$\Phi(\mathbf{x})$ comes from the background density $\rho_{\hbox{\scriptsize{asymp}}}$. Thus, a good fraction of the mass in the observable galaxy *does not contribute to the motion of test particles in the galaxy*. It is rather the near-core density $\rho_{II}^1(r)$ that contributes to $\mathfrak{V}(\mathbf{x})$. As inferring the mass of structures through observations of the dynamics under gravity of their constituents is one of the main ways of estimating mass, the motion of stars in galaxies can only be used to estimate $\rho_{II}^1$; the matter in $\rho_{\hbox{\scriptsize{asymp}}}(r)$ is present, but cannot be “seen” in this way. Moreover, as $\rho_{\hbox{\scriptsize{asymp}}}(r)\gg \rho_{II}^1(r)$ when $r\gg
r_H$, *the majority of the mass in the universe cannot be seen using these methods*.
A Cosmological Check
====================
We have extrapolated our results for a single galaxy to the cosmological scale. This is possible because recent measurements from WMAP, the Supernova Legacy Survey, and the HST key project show that the universe is essentially flat; $h=0.732_{-0.032}^{+0.031}$ and of the age of the universe $t_0=13.73_{-0.15}^{+0.16}$ Gyr were determined using this assumption. The largest distance between galaxies is thus $ct_0\equiv \mathfrak{K}(\Omega)
\lambda_{H}$, where $\mathfrak{K}(\Omega) =1.03_{\pm0.05}$.
Next, the density of matter of our model galaxy dies off exponentially fast at $r_{II}$; the extent of matter in the galaxy is fundamentally limited to $2r_{II}$. This size does not depend on the detailed structure of the galaxy; it is inherent to the theory. Given a $\Omega_\Lambda = 0.716_{\pm
0.055}$, we can express $r_{II}
=[8\pi\chi/3\Omega_\Lambda(1+4^{1+\alpha_\Lambda})]^{1/2}\lambda_H$ [@ADS] as well [@WMAP], and numerically $r_{II}=0.52\lambda_H$ for $\alpha_\Lambda = 3/2$. Although $\alpha_\Lambda$ was set to $3/2$ based on analysis at the galactic scale, $\rho(r)$ naturally cuts off at $\lambda_H/2$.
To accomplish the extrapolation, we consider our model galaxy to b
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turbation theory if the superdimension of the representation of the primary is non-zero (i.e. for short multiplets). For example for the short, discrete representation crucial to the calculation in [@Ashok:2009jw], there are no corrections.
Notice that the stress-energy tensor can also be written in terms of the right currents. Equation implies that a primary field transforms under the left- and right-action of the group in representations that have the same eigenvalue of the quadratic Casimir operator. The simple poles in and also give the relations: \[dPhi=JPhi\] (z) = \_[ba]{}t\^a :j\^b\_[L,z]{}:(z)+ (f\^4) \[dbarPhi=JbarPhi\] | (z) = \_[ba]{}t\^a :j\^b\_[L,|z]{}:(z)+ (f\^4).
### Remark about the atypical sector {#remark-about-the-atypical-sector .unnumbered}
Some of the primary fields are associated to atypical Kac modules, that are reducible but indecomposable [@Gotz:2006qp]. In that case the matrices $t^a$ that appear in equation are not invertible. Moreover the quadratic operator $\kappa_{ba}t^a
t^b$ can then be written in an upper-triangular form, with zeros on the diagonal (which is the generalized eigenvalue of the quadratic casimir for atypical representations of e.g. the $psl(n|n)$ superalgebra). Equation tells us that the operator $L_0$ is proportional to this quadratic operator $\kappa_{ba}t^a t^b$ when acting on a primary field. This implies that $L_0$ is non-diagonalizable, which betrays the logarithmic nature of the theory (see [@Gotz:2006qp] for a similar argument in the case of $psl(2|2)$, and [@Gaberdiel:2001tr],[@Flohr:2001zs] for an introduction to logarithmic CFTs). Let us remark here that the fact that the current component $j_{L,z}$ has dimensions $(1,0)$, but is not holomorphic also codes the logarithmic nature of the conformal field theory [@Read:2001pz].
A recursive bootstrap for the elementary operator algebra {#bootstrap}
==========================================================
In this section we will explain how to compute the current-current and current-primary OPEs order b
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uspin-DeltaACP.bib'
title: 'The Emergence of the $\Delta U=0$ Rule in Charm Physics'
---
Introduction \[sec:intro\]
==========================
In a recent spectacular result, LHCb discovered direct CP violation in charm decays at 5.3$\sigma$ [@Aaij:2019kcg]. The new world average of the difference of CP asymmetries [@Aitala:1997ff; @Link:2000aw; @Csorna:2001ww; @Aubert:2007if; @Staric:2008rx; @Aaltonen:2011se; @Collaboration:2012qw; @Aaij:2011in; @Aaij:2013bra; @Aaij:2014gsa; @Aaij:2016cfh; @Aaij:2016dfb] $$\begin{aligned}
\Delta a_{CP}^{\mathrm{dir}} &\equiv
a_{CP}^{\mathrm{dir}}(D^0\rightarrow K^+K^-) - a_{CP}^{\mathrm{dir}}(D^0\rightarrow \pi^+\pi^-)\,, \end{aligned}$$ where $$\begin{aligned}
a_{CP}^{\mathrm{dir}}(f) &\equiv \frac{
\vert \mathcal{A} (D^0\to f)\vert^2 - \vert {\mathcal{A}}(\overline{D}^0\to f)\vert^2
}{
\vert \mathcal{A}(D^0\to f)\vert^2 + \vert {\mathcal{A}}(\overline{D}^0\to f)\vert^2
}\,, \end{aligned}$$ and which is provided by the Heavy Flavor Averaging Group (HFLAV) [@Amhis:2016xyh], is given as [@Carbone:2019] $$\begin{aligned}
\Delta a_{CP}^{\mathrm{dir}} &= -0.00164\pm 0.00028\,. \label{eq:HFLAVav} \end{aligned}$$ Our aim in this paper is to study the implications of this result. In particular, working within the Standard Model (SM) and using the known values of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements as input, we see how Eq. (\[eq:HFLAVav\]) can be employed in order to extract low energy QCD quantities, and learn from them about QCD.
The new measurement allows for the first time to determine the CKM-suppressed amplitude of singly-Cabibbo-suppressed (SCS) charm decays that contribute a weak phase difference relative to the CKM-leading part, which leads to a non-vanishing CP asymmetry. More specifically, $\Delta a_{CP}^{\mathrm{dir}}$ allows to determine the imaginary part of the $\Delta U=0$ over $\Delta U=1$ matrix elements.
As we show, the data suggest the emergence of a $\Delta U=0$ rule, which has features that are similar to the known $\Delta I=1/2$ rule in
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cal{U}}(\chi )_{{\alpha }_i}$, and $F_i\in {\mathcal{U}}(\chi )_{-{\alpha }_i}$ for all $i\in I$. Let $${\mathbb{N}}_0^I=\Big\{\sum _{i\in I}a_i{\alpha }_i\,|\,a_i\in {\mathbb{N}}_0\Big\}\subset
{\mathbb{Z}}^I,$$ and for any subspace ${\mathcal{U}}'\subset {\mathcal{U}}(\chi )$ and any $\beta \in {\mathbb{Z}}^I$ let ${\mathcal{U}}'_\beta ={\mathcal{U}}'\cap {\mathcal{U}}(\chi )_\beta $. Then $$\begin{aligned}
{\mathcal{U}}^+(\chi )=&\oplus _{\beta \in {\mathbb{N}}_0^I}{\mathcal{U}}^+(\chi )_\beta ,&
{\mathcal{U}}^-(\chi )=&\oplus _{\beta \in {\mathbb{N}}_0^I}{\mathcal{U}}^-(\chi )_{-\beta }.\end{aligned}$$
For all $\beta \in {\mathbb{Z}}^I$ let $$\begin{aligned}
|\beta |=\sum _{i\in I} a_i\in {\mathbb{Z}},\quad \text{ where }
\beta =\sum _{i\in I}a_i{\alpha }_i.
\label{eq:abs}\end{aligned}$$ The decomposition $${\mathcal{U}}(\chi )=\oplus _{m\in {\mathbb{Z}}}{\mathcal{U}}(\chi )_m,\quad
\text{where}\quad
{\mathcal{U}}(\chi )_m=\oplus _{\beta :|\beta |=m}{\mathcal{U}}(\chi )_\beta ,
\label{eq:Zgrading}$$ gives a ${\mathbb{Z}}$-grading of ${\mathcal{U}}(\chi )$ called the *standard grading*.
\[pr:algiso\] Let $\chi \in {\mathcal{X}}$.
\(1) Let ${\underline{a}}=(a_i\,|\,i\in I)\in ({{\Bbbk }^\times })^I$. Then there exists a unique algebra automorphism $\varphi _{{\underline{a}}}$ of ${\mathcal{U}}(\chi )$ such that $$\varphi _{{\underline{a}}}(K_i)=K_i,\,\,
\varphi _{{\underline{a}}}(L_i)=L_i,\,\,
\varphi _{{\underline{a}}}(E_i)=a_iE_i,\,\,
\varphi _{{\underline{a}}}(F_i)=a_i^{-1}F_i.
\label{eq:cUauto1}$$
\(2) There is a unique algebra antiautomorphism ${\Omega }$ of ${\mathcal{U}}(\chi )$ such that $$\begin{aligned}
{\Omega }(K_i)=&K_i,& {\Omega }(L_i)=&L_i,&
{\Omega }(E_i)=&F_i,& {\Omega }(F_i)=&E_i.
\label{eq:cUantiauto}\end{aligned}$$ It satisfies the relation ${\Omega }^2={\operatorname{id}}$.
\[le:commEFi\] For all $i\in I$ there exist unique linear maps ${\partial ^K}_i,{\partial ^L}_i\in {\mathrm{End}}_{\Bbbk }({\mathcal{U}}^+(\chi ))$ such that $$\begin
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{\mbox{\boldmath $\alpha$}})
= \oplus_{P \in {\mathbb T}({\mbox{\boldmath $\alpha$}}) }K_0 v_P$ be a vector space over $K_0$ with the standard basis $\{v_P|P\in {\mathbb T}({\mbox{\boldmath $\alpha$}})\}$.
For generators $e_i$, $f_i$ and $s_i$ of ${A}_n$, we define linear maps $\rho_{{\mbox{\boldmath $\alpha$}}}(e_i)$, $\rho_{{\mbox{\boldmath $\alpha$}}}(f_i)$ and $\rho_{{\mbox{\boldmath $\alpha$}}}(s_i)$ on ${\mathbb V}({\mbox{\boldmath $\alpha$}})$ giving the matrices $E_i$ $F_i$ and $M_i$ respectively with respect to the basis $\{ v_P | P\in {\mathbb T}({\mbox{\boldmath $\alpha$}}) \}$.
Definition of $\rho_{{\mbox{\boldmath $\alpha$}}}(e_i)$
-------------------------------------------------------
Firstly, we define a linear map for $e_i$.
For a tableaux $P = ({\mbox{\boldmath $\alpha$}}^{(0)}, {\mbox{\boldmath $\alpha$}}^{(1/2)}, \ldots, {\mbox{\boldmath $\alpha$}}^{(n)})$ of ${\mathbb T}({\mbox{\boldmath $\alpha$}})$, we define $\rho_{{\mbox{\boldmath $\alpha$}}}(e_i)(v_P)
= \sum_{Q \in {\mathbb T}({\mbox{\boldmath $\alpha$}})}(E_i)_{QP}v_Q$. Let $Q = ({\mbox{\boldmath $\alpha$}}^{\prime(0)}, {\mbox{\boldmath $\alpha$}}^{\prime(1/2)},
\ldots, {\mbox{\boldmath $\alpha$}}^{\prime(n)})$.
If there is an $i_0 \in \{1/2, 1, \ldots, n-1/2 \} \setminus \{i-1/2\}$ such that ${\mbox{\boldmath $\alpha$}}^{(i_0)}\neq {\mbox{\boldmath $\alpha$}}^{\prime(i_0)}$, then we put $$(E_i)_{QP} = 0.$$ In the following, we consider the case that ${\mbox{\boldmath $\alpha$}}^{(i_0)} = {\mbox{\boldmath $\alpha$}}^{\prime(i_0)}$ for $i_0\in\{0, 1/2, 1, \ldots, n-1/2\}\setminus\{i-1/2\}$.
If ${\mbox{\boldmath $\alpha$}}^{(i-1)}$ and ${\mbox{\boldmath $\alpha$}}^{(i)}$ are not labeled by the same Young diagram, then we put $$(E_i)_{QP} = 0.$$
We consider the case ${\mbox{\boldmath $\alpha$}}^{(i-1)}$ and ${\mbox{\boldmath $\alpha$}}^{(i)}$ have the same label $\widetilde{\lambda}$. In this case, the possible vertices as ${\mbox{\boldmath $\alpha$}}^{(i-1/2)}$ have labels $\{\widetilde{\lambda}^{-}_{(s)}\}$, which are obtained by
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}A\quad & \textrm{Equation}\!:\textrm{ }{\textstyle {\mu\Phi_{x}+\Phi=(c/2)\int_{-1}^{1}\Phi(x,\mu^{\prime})d\mu^{\prime},\; x\geq0}}\\
& \textrm{Boundary condition}:\textrm{ }\Phi(0,\mu)=0,\;\mu\geq0\\
& \textrm{Asymptotic condition}:\textrm{ }\Phi\rightarrow e^{-x/\nu_{0}}\phi(\mu,\nu_{0}),\; x\rightarrow\infty.\\
Problem\textrm{ }B\quad & \textrm{Equation}\!:\textrm{ }{\textstyle {\mu\Phi_{x}+\Phi=(c/2)\int_{-1}^{1}\Phi(x,\mu^{\prime})d\mu^{\prime},\; x\geq0}}\\
& \textrm{Boundary condition}:\textrm{ }\Phi(0,\mu)=1,\;\mu\geq0\\
& \textrm{Asymptotic condition}:\textrm{ }\Phi\rightarrow0,\; x\rightarrow\infty.\end{aligned}$$
The full $-1\leq\mu\leq1$ range form of the half $0\leq\mu\leq1$ range discretized spectral approximation replaces the exact integral boundary condition at $x=0$ by a suitable quadrature sum over the values of $\nu$ taken at the zeros of Legendre polynomials; thus the condition at $x=0$ can be expressed as $$\psi(\mu)=a(\nu_{0})\phi(\mu,\nu_{0})+\sum_{i=1}^{N}a(\nu_{i})\phi_{\varepsilon}(\mu,\nu_{i}),\qquad\mu\in[0,1],\label{Eqn: BC}$$
where $\psi(\mu)=\Phi(0,\mu)$ is the specified incoming radiation incident on the boundary from the left, and the half-range coefficients $a(\nu_{0})$, $\{ a(\nu)\}_{\nu\in[0,1]}$ are to be evaluated using the $W$-function of Appendix 4. We now exploit the relative simplicity of the full-range calculations by replacing Eq. (\[Eqn: BC\]) by Eq. (\[Eqn: HRFR\_Discrete\]) following, where the coefficients $\{ b(\nu_{i})\}_{i=0}^{N}$ are used to distinguish the full-range coefficients from the half-range ones. The significance of this change lies in the overwhelming simplicity of the full-range weight function $\mu$ as compared to the half-range function $W(\mu)$, and the resulting simplicity of the orthogonality relations that follow, see Appendix A4. The basic data of $z_{0}$ and $X(-\nu)$ are then completely generated self-consistently [@Sengupta1988; @Sengupta1995] by the discretized spectral approximation from the full-range adaption $$\sum_{i=0}^{N}b_{
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plest examples would be $$\begin{aligned}
{\label{eq:Juniform-def}}
J_{o,x}=\frac{{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{0<\|x\|_\infty\leq L\}$}}}}{\sum_{z\in{{\mathbb Z}^d}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{0<\|z
\|_\infty\leq L\}$}}}}=O(L^{-d})\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{0<\|L^{-1}x\|_\infty\leq1\}$}}}.\end{aligned}$$
\[prp:Pij-Rj-bd\] Let $\rho=2(d-4)>0$. For the nearest-neighbor model with $d\gg1$ and for the spread-out model with $L\gg1$, there are finite constants $\theta$ and $\lambda$ such that $$\begin{aligned}
{\label{eq:prp-bds}}
|\Pi_{p;\Lambda}^{{\scriptscriptstyle}(j)}(x)-\delta_{o,x}|&\leq\theta\delta_{o,x}
+\frac{\lambda(1-\delta_{o,x})}{|x|^{d+2+\rho}}\quad(j\ge0),&&
|R_{p;\Lambda}^{{\scriptscriptstyle}(j)}(x)|\to0\quad(j\uparrow\infty),\end{aligned}$$ for any $p\leq{p_\text{c}}$, any $\Lambda\subset{{\mathbb Z}^d}$ and any $x\in\Lambda$.
The proof of Proposition \[prp:Pij-Rj-bd\] depends on certain bounds on the expansion coefficients in terms of two-point functions. These diagrammatic bounds arise from counting the number of “disjoint connections”, corresponding to applications of the BK inequality in percolation (e.g., [@bk85]). We prove these bounds in Section \[s:bounds\], and in anticipation of this, in Section \[s:reduction\] we explain how we use their implication to prove Proposition \[prp:Pij-Rj-bd\], with $\theta=O(d^{-1})$ and $\lambda=O(1)$ for the nearest-neighbor model, and $\theta=O(L^{-2+{\epsilon}})$ and $\lambda=O(\theta^2)$ with a small ${\epsilon}>0$ for the spread-out model.
Let $$\begin{aligned}
\tau\equiv\tau(p)=\sum_x\tau_{o,x},&&
D(x)=\frac{\tau_{o,x}}{\tau},&&
\sigma^2=\sum_x|x|^2D(x).\end{aligned}$$ Due to [(\[eq:prp-bds\])]{} uniformly in $\Lambda\subset{{\mathbb Z}^d}$, there is a limit $\Pi_p(x)\equiv\lim_{\Lambda\uparrow{{\mathbb Z}^d}}
\lim_{j\uparrow\infty}\Pi_{p;\Lambda}^{{\scriptscriptstyle}(j)}(x)$ such that $$\begin{aligned}
{\label{eq:Ising-lace-Zdlim}}
G_p(x)=\Pi_p(x)+(\Pi_p*\tau D*G_p)(x),&&
|\Pi_p(x)-\delt
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the intrinsic fluctuations of the outflow rates present in the quasi-steady states.
![The net accretion rate $\dot{M}$ normalized by the Bondi rate $\dot{M}_{\mathrm{B}}$ as a function of the opening angle of the horizontal neutral layer at the Bondi radius $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$. The crosses show the runs with shadow opening angles $\theta_{\mathrm{shadow}}=45^\circ$, $33.75^\circ$, $22.5^\circ$ and $11.25^\circ$. The solid line represents the relation given by equation assuming $\dot{M}_{\mathrm{loss}}=0.07\dot{M}_{\mathrm{B}}$. []{data-label="fig:mdot_th_indep"}](figure/mdot_th_indep.eps){width="8.5cm"}
The net accretion rates $\dot{M}$ are plotted as crosses in Fig. \[fig:mdot\_th\_indep\] against $\theta_{\mathrm{inflow}}(r_{\mathrm{B}})$. Since equation slightly overestimates $\dot{M}_{\mathrm{inflow}}(R_{\mathrm{in}})$ due to the photoevaporation mass loss, we modify equation assuming a constant mass loss rate $\dot{M}_{\mathrm{loss}}$ in all cases, as $$\begin{aligned}
\dot{M} = \dot{M}_{\mathrm{inflow}}(R_{\mathrm{in}}) =
\frac{\Delta\Omega_{\mathrm{inflow}}(r_{\mathrm{B}})}{4\pi}\dot{M}_{\mathrm{B}}
- \dot{M}_{\mathrm{loss}} \,.
\label{eq:18}\end{aligned}$$ We find that $\dot{M}_{\mathrm{loss}}=0.07\,\dot{M}_{\mathrm{B}}$ gives the best fit to the simulated results with errors less than 2% of $\dot{M}_{\mathrm{B}}$. This good agreement also supports the above assumption of constant $\dot{M}_{\mathrm{loss}}$. The value of $\dot{M}_{\mathrm{loss}}$ is similar to but smaller than $\dot{M}_{\mathrm{outflow}}(r_{\mathrm{B}})$ (equation \[eq:14\]) partly due to the contribution from the circulation flows, as mentioned in Sec. \[sec:anl\_modeling\]. Moreover, by setting $\dot{M} = 0$ in equation , we get the critical opening angle $\theta_{\mathrm{cr}} \simeq 4^\circ$, below which the equatorial neutral flow disappears by photoevaporation. This value will be raised up to $\theta_{\mathrm{cr}} \simeq 10^\circ$ if we include the mass loss inside the sink, which is currently ignor
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i \setminus \Phi][\Phi]$ is well-defined. By definition \[D:DYADIC\_ENSEMBLE\_PRODUCT\], $[\Psi \setminus \Phi][\Phi] = (\Psi \setminus \Phi) \cup \Phi = \Psi$.
The case $\Phi = \varnothing$ does not occur naturally in systems theory because no proper system is unresponsive to all possible stimuli. When $\Psi = \Phi$, the basis has no event space through which to receive transient external stimuli.
Uncoverable processes {#S:UNCOVERABLE_PROCESS_APPENDIX}
=====================
Although any procedure does cover some process, some processes have no covering procedure. This disparity arises naturally through limiting the quantity of distinct functionalities participating in a procedure. Here the constraining mechanism is the catalog of functionality, whose membership must be finite. This stricture’s rationale is to emulate software, which is presumed to possess finite functionality.
Uncoverability of a process entails more than failure of definition \[D:COVERING\_PROCEDURE\] in the case of a particular procedure; uncoverability implies failure for *any* procedure constructed from a given catalog of functionality. With process $\lbrace {\mathbf{f}}_n \rbrace \colon {\mathbb{N}}\to {\prod{\Psi}} \times {\prod{\Phi}}$ and catalog of functionality ${\mathscr{F}}$, uncoverability requires an $i \in {\mathbb{N}}$ and term ${\mathbf{f}}_i = {\mathbf{f}}$ such that ${\mathbf{f}} \notin {\mathit{f}}$ for each ${\mathit{f}} \in {\mathscr{F}}$.
Pigeonhole principle
--------------------
The pigeonhole principle can verify uncoverability, but not coverability. Suppose two frames have the same abscissa but different ordinates. No single functionality can cover both frames, since functionalities are mappings. In more general analogy, let distinct equi-abscissa frames be pigeons, while functionalities be pigeonholes. If more than $N$ pigeons occupy $N$ pigeonholes, then some pigeonhole contains more than one pigeon, which is not allowed.
Underpigeonholing
-----------------
The *frame set* derived from the initial segment of
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Therefore, using the fact that $(e_i - e_{\i})(e_i - e_{\i})^\top$ is positive semi-definite, and Equations , and we have $$\begin{aligned}
\label{eq:topl_expec}
\E[M] &\succeq& e^{-2b} \sum_{j = 1}^n \frac{\ell_j}{\kappa_j(\kappa_j-1)} \sum_{i<\i \in S_j} (e_i - e_{\i})(e_i - e_{\i})^\top = e^{-2b} L,\end{aligned}$$ where $L$ is the Laplacian defined for the comparison graph $\H$, Definition \[def:comparison\_graph1\]. Using $\lambda_2(L) = (\alpha/(d-1))\sum_{j = 1}^n \ell_j$ from , we get the desired bound $\lambda_2(\E[M]) \geq e^{-2b}(\alpha/(d-1))\sum_{j = 1}^n \ell_j$.
For top-$\ell_j$ rank breaking, $M^{(j)}$ is also given by $$\begin{aligned}
\label{eq:topl7}
M^{(j)} = \frac{1}{\kappa_j -1}\Big((\kappa_j - \ell_j)\diag(e_{\{I_j\}}) +\ell_j \diag(e_{\{S_j\}}) - e_{\{I_j\}}e_{\{S_j\}}^\top - e_{\{S_j\}}e_{\{I_j\}}^\top + e_{\{I_j\}}e_{\{I_j\}}^\top \Big),\end{aligned}$$ where $e_{\{S_j\}},e_{\{I_j\}} \in \reals^d$ are zero-one vectors, $e_{\{S_j\}}$ has support corresponding to the set of items $S_j$ and $e_{\{I_j\}}$ has support corresponding to the random top-$\ell_j$ items in the ranking $\sigma_j$. $I_j = \{\sigma_j(1), \sigma_j(2),\cdots, \sigma_j(\ell_j)\}$ for $j \in [n]$. $(M^{(j)})^2$ is given by $$\begin{aligned}
(M^{(j)})^2 &=& \frac{1}{(\kappa_j -1)^2}\Big((\kappa_j^2 - \ell_j^2)\diag(e_{\{I_j\}}) + {\ell_j}^2\diag(e_{\{S_j\}}) - \nonumber\\
&& \hspace{5em}(\kappa_j +\ell_j)(e_{\{I_j\}}e_{\{S_j\}}^\top + e_{\{S_j\}}e_{\{I_j\}}^\top -e_{\{I_j\}}e_{\{I_j\}}^\top ) + \ell_j e_{\{S_j\}}e_{\{S_j\}}^\top \Big).\end{aligned}$$ Note that $\P[i \in I_j| i \in S_j] \leq \ell_j e^{2b}/\kappa_j$ for all $i \in S_j$. Its proof is similar to the proof of Lemma \[lem:prob\_toplbound\]. Therefore, we have $\E[\diag(e_{\{I_j\}})] \preceq \ell_j e^{2b}/\kappa_j \diag(e_{\{{\boldsymbol{1}}\}})$. To bound $\|\sum_{j =1}^n\E[(M^{(j)})^2]\|$, we use the fact that for $J \in \reals^{d\times d}, {\|J\|} \leq \max_{i \in [d]}\sum_{\i = 1}^d|J_{i\i}|$. Maximum of row sums of $\E[e_{\{I_j\}}e_{\{I_j\}}^\top]$ is up
| 2,811
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roject2).
I am also taking AI this semester and was considering expanding upon something
like this for my masters' capstone project. The application of machine
learning to this type of situation is going to be very interesting to say the
least.
Just wanted to say good luck!
------
juskrey
Basically optimizations and "modelling" are the reason resource allocations
are total failure in the case of rare emergencies, like ongoing. You assume
the worst "one in a 1000 years" case, multiply by 2..10, depending on your
cash flow, and keep allocations up to date. Period.
And the simple truth again: you can't immediately allocate during a crisis.
------
coderthrow
I am modelling with American Community Survey complete raw data, physically in
Berkeley, California.. using PostGIS, python and an SEIR model; Urban Planning
background.. suggestions welcome
~~~
coderthrow
example output, US Persons by Age-Sex 50+ by Census Tract, nationwide.. exec.
time 1230 ms. local, no clouds
-[ RECORD 221408 ]----+
mtable_2_pkey | 18744349
geoid | 08000US361198400084000000900
geo_name | Census Tract 9, Yonkers city,
Yonkers city, Westchester County, New York
Total_Population | 2307
Male | 1085
male over 50 est. | 285
Female | 1222
female over 50 est. | 272
-[ RECORD 221409 ]----+
mtable_2_pkey | 18754394
geoid | 08000US421338704887048000900
geo_name | Census Tract 9, York city, York city, York County, Pennsylvania
Total_Population | 7100
Male | 3934
male over 50 est. | 1453
Female | 3166
female over 50 est. | 1347
-[ RECORD 221410 ]----+
mtable_2_pkey | 18776479
geoid | 14000US42133000900
geo_name | Census Tract 9, York County,
Pennsylvania
Total_Population | 11
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$i\in I$, $a\in A$.
3. ${\sigma }_i^a(R^a) = R^{{r}_i(a)}$ for all $i\in I$, $a\in A$.
4. If $i,j\in I$ and $a\in A$ such that $i\not=j$ and $m_{i,j}^a$ is finite, then $({r}_i{r}_j)^{m_{i,j}^a}(a)=a$.
If ${\mathcal{R}}$ is a root system of type ${\mathcal{C}}$, then ${\mathcal{W}}({\mathcal{R}})={\mathcal{W}}({\mathcal{C}})$ is the *Weyl groupoid of* ${\mathcal{R}}$. Further, ${\mathcal{R}}$ is called *connected*, if ${\mathcal{C}}$ is a connected Cartan scheme. If ${\mathcal{R}}={\mathcal{R}}({\mathcal{C}},(R^a)_{a\in A})$ is a root system of type ${\mathcal{C}}$ and ${\mathcal{R}}'={\mathcal{R}}'({\mathcal{C}}',({R'}^a_{a\in A'}))$ is a root system of type ${\mathcal{C}}'$, then we say that ${\mathcal{R}}$ and ${\mathcal{R}}'$ are *equivalent*, if ${\mathcal{C}}$ and ${\mathcal{C}}'$ are equivalent Cartan schemes given by maps $\varphi
_0:I\to I'$, $\varphi _1:A\to A'$ as in Def. \[de:CS\], and if the map $\varphi _0^*:\ndZ^I\to \ndZ^{I'}$ given by $\varphi _0^*({\alpha }_i)={\alpha }_{\varphi _0(i)}$ satisfies $\varphi _0^*(R^a)={R'}^{\varphi _1(a)}$ for all $a\in A$.
There exist many interesting examples of root systems of type ${\mathcal{C}}$ related to semisimple Lie algebras, Lie superalgebras and Nichols algebras of diagonal type, respectively. For further details and results we refer to [@a-HeckYam08] and [@p-CH08].
\[con:uind\] In connection with Cartan schemes ${\mathcal{C}}$, upper indices usually refer to elements of $A$. Often, these indices will be omitted if they are uniquely determined by the context. In particular, for any $w,w'\in {\mathrm{Hom}}({\mathcal{W}}({\mathcal{C}}))$ and $a\in A$, the notation $1_aw$ and $w'1_a$ means that $w\in {\mathrm{Hom}}(\underline{\,\,},a)$ and $w'\in {\mathrm{Hom}}(a,\underline{\,\,})$, respectively.
A fundamental result about Weyl groupoids is the following theorem.
[@a-HeckYam08 Thm.1]\[th:Coxgr\] Let ${\mathcal{C}}={\mathcal{C}}(I,A,({r}_i)_{i\in I},(C^a)_{a\in A})$ be a Cartan scheme and ${\mathcal{R}}={\mathcal{R}}({\mathcal{C}},(R^a)_{a\in A
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c_\gamma e^{-\gamma}$ with $\Gamma_G \subseteq \Lambda^*_G$ finite and all $c_\gamma \neq 0$. Then, $$m_{G,V}(\lambda) = \sum_{\gamma \in \Gamma_G} c_\gamma \, m_{T_G,V}(\lambda + \gamma).$$
In particular, it is evident from that, for any fixed group $G$, the multiplicity of an irreducible representation in some representation $V$ can be computed efficiently from the weight multiplicities of $V$ by computing a finite linear combination.
Multiplicities for the Subgroup Restriction Problem {#section:finite difference formula for subgroup restrictions}
===================================================
Every $G$-representation $V$ can be considered as (“restricts to”) a representation of $H$ by setting $$\label{restriction}
h \cdot v := f(h) \cdot v \qquad (\forall h \in H),$$ and the subgroup restriction problem for $f$, as defined in , amounts to determining the multiplicity $m^\lambda_\mu$ of a given irreducible representation of $H$ in the restriction of a given irreducible representation of $G$. In this section we will derive a formula for these multiplicities (), which will be the main ingredient of the algorithm presented in below. It will also follow from this formula that the $m^\lambda_\mu$ are given by a piecewise quasi-polynomial function[^2] in $\lambda$ and $\mu$ ().
Let us choose the maximal torus $T_H \subseteq H$ in such a way that $f(T_H) \subseteq T_G$, and denote the corresponding Cartan subalgebra by $\mathfrak t_H$. Of course, this implies that the induced Lie algebra homomorphism $\operatorname{Lie}(f)$ sends the Cartan subalgebra of $H$ in the one of $G$. Since $f$ is a group homomorphism, $\operatorname{Lie}(f)$ restricts to a homomorphism between the integral lattices, $F \colon \Lambda_H \rightarrow \Lambda_G, ~ X \mapsto \operatorname{Lie}(f) X$. The dual map between the weight lattices is given by $$\label{definition dual map}
F^* \colon \Lambda_G^* \rightarrow \Lambda_H^*, \quad
\beta \mapsto \beta \circ F = {\left.\beta \circ \operatorname{Lie}(f)\vphantom{\big|}\right|_
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- Q^TG^{-1} Q + e^T e \ ,$$ where we introduce the frame fields $$G= \kappa^T \kappa \ , \quad e= \kappa \left( L + G^{-1} Q \right) .$$
We perform the dualisation by introducing a $\mathfrak{h}$-valued connection with components $A_{\pm} = A_{\pm}^{a}H_{a}$ and a $\mathfrak{h}^\star$-valued Lagrange multiplier $V= v_{a} \tilde{H}^{a}$. We covariantise currents $$J^{\nabla}_{\pm} = g^{-1} d g + g^{-1} A_{\pm } g \ ,$$ such that we are gauging a left action of some $\tilde{h} \in H$ $$g \rightarrow \tilde{h} g \ , \quad A \rightarrow \tilde{h} A \tilde{h} ^{-1} - d \tilde{h} \tilde{h}^{-1} \ ,$$ and consider $${\cal L }^{ \nabla} = {\operatorname{Tr}}(J^{\nabla}_+ P(J^{\nabla}_-) ) + {\operatorname{Tr}}(V F_{+ -} ) \ ,$$ where the field strength is $F_{+-} = \partial_{+} A_{-} - \partial_{-} A_{+} + [A_{+} , A_{-}]$.
We continue by gauge fixing on the group element $g = \hat{g}$ i.e. $h=1$.[^2] Integrating the Lagrange multipliers enforces a flat connection and one recovers the starting model since $$\label{eq:puregauge}
A_\pm = h^{-1}\partial_\pm h = L_\pm \ ,$$ and upon substituting back into the action one recovers the starting $\sigma$-model.
On the other hand, integrating by parts the derivative terms of the gauge fields yields $${\cal L }^{ \nabla} = {\operatorname{Tr}}(\hat{J}_{+} P(\hat{J}_-) ) + A_{+}^{a}A_{-}^{b} M_{ab} + A_{+}^{a}( \partial_{-} v_{a}+ Q_{-a} ) - A_{-}^{a}( \partial_{+} v_{a} - Q_{+a} ) \ ,$$ in which we have pulled back the one-forms $Q$ and $\hat{J}$ to the worldsheet and defined $$\begin{aligned}
F_{ab} &= {\operatorname{Tr}}([H_{a} ,H_{b}]V) = f_{ab}{}^{c} v_{c} \ , \quad M_{ab} =G_{ab} + F_{ab} \ .
\end{aligned}$$ The gauge field equations of motion now read $$\label{eq:gauge}
A_{-} = - M^{-1} ( \partial_{-} v + Q_{-} ) \ , \quad A_{+} = M^{-T} ( \partial_{+} v - Q_{+ } ) \ .$$ Combining these equations of motion for the gauge field in eqs. and sets up the canonical transformation between T-dual theories. Substitution of the gauge field equation of motion into the action yields
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athbf{b}_j^T\mathbf{BX}^\dagger\bigg)^T$$ $$=\frac{1}{\lambda_i\lambda_j}\mathbf{b}_i^T(\mathbf{BX}^\dagger)(\mathbf{BX}^\dagger)^T\mathbf{b}_j=\frac{1}{\lambda_i\lambda_j}\mathbf{b}_i^T\mathbf{B}\mathbf{b}_j$$ $$=\frac{1}{\lambda_i}\mathbf{b}_i^T\mathbf{b}_j=0,$$ so $$\mathbf{c}_i^T\mathbf{c}_j=\frac{\mathbf{m}_i^T\mathbf{m}_j}{\|\mathbf{m}_i\|_2\|\mathbf{m}_j\|_2}$$ implies $\mathbf{c}_i\perp\mathbf{c}_j$ for $i\neq j$.
From Theorem \[thm5\], it can be seen that the modularity components are orthogonal to each other. Next we prove that the projection of the uncentered data onto the span of $\mathbf{c}_i$ is a scalar multiple of $\mathbf{b}_i$.
\[thm6\] With the assumptions in Lemma \[thm3\], let $\mathbf{P}_{\mathbf{c}_i}$ be the projector onto the span of $\mathbf{c}_i$. Then we have $$\mathbf{P}_{\mathbf{c}_i}\mathbf{X}=\frac{1}{\|\mathbf{m}_i\|_2}\mathbf{c}_i\mathbf{b}_i^T.$$
$$\mathbf{P}_{\mathbf{c}_i}\mathbf{X}=\mathbf{c}_i\mathbf{c}_i^T\mathbf{X}=\frac{1}{\|\mathbf{m}_i\|_2}\mathbf{c}_i\mathbf{m}_i^T\mathbf{U\Sigma V}^T$$ $$=\frac{1}{\|\mathbf{m}_i\|_2}\mathbf{c}_i\bigg(\sum_{j=1}^k\frac{\gamma_{ij}}{\sigma_j}\mathbf{u}_j^T\bigg)\mathbf{U\Sigma V}^T$$ $$=\frac{1}{\|\mathbf{m}_i\|_2}\mathbf{c}_i\begin{pmatrix}
\frac{\gamma_{i1}}{\sigma_1} & \frac{\gamma_{i2}}{\sigma_2} & \cdots & \frac{\gamma_{ik}}{\sigma_k} & 0 & \cdots & 0
\end{pmatrix}_{1\times p}\Sigma\mathbf{V}^T$$ $$=\frac{1}{\|\mathbf{m}_i\|_2}\mathbf{c}_i\begin{pmatrix}
\gamma_{i1} & \gamma_{i2} & \cdots &\gamma_{ik} & 0 & \cdots & 0
\end{pmatrix}_{1\times n}\mathbf{V}^T$$ $$=\frac{1}{\|\mathbf{m}_i\|_2}\mathbf{c}_i\sum_{j=1}^k\gamma_{ij}\mathbf{v}_i^T=\frac{1}{\|\mathbf{m}_i\|_2}\mathbf{c}_i\mathbf{b}_i^T.$$
This property is similar to that of principal components in the sense that if we project the data onto the span of the components, we get a scalar multiple of a vector, and the vector can give the clusters in the data based on the signs of the entries in the eigenvectors. Finally, we can prove that if we look at $\mathbf{X}$ in the space p
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owth condition]{}**]{}).
This is the basis of the following characterization theorem. For the proof we refer to [@PS91; @Kon80; @HKPS93; @KLPSW96].
\[charthm\] A mapping $F:S_{d}({\mathbb{R}}) \to {\mathbb{C}}$ is the $T$-transform of an element in $(S)'$ if and only if it is a U-functional.
Theorem \[charthm\] enables us to discuss convergence of sequences of Hida distributions by considering the corresponding $T$-transforms, i.e. by considering convergence on the level of U-functionals. The following corollary is proved in [@PS91; @HKPS93; @KLPSW96].
\[seqcor\] Let $(\Phi_n)_{n\in {\mathbb{N}}}$ denote a sequence in $(S)'$ such that:
- For all ${\bf f} \in S_{d}({\mathbb{R}})$, $((T\Phi_n)({\bf f}))_{n\in {\mathbb{N}}}$ is a Cauchy sequence in ${\mathbb{C}}$.
- There exist constants $0<C,D<\infty$ such that for some $p \in {\mathbb{N}}_0$ one has $$|(T\Phi_n)(z{\bf f })|\leq C\exp(D|z|^2\|{\bf f}\|_p^2)$$ for all ${\bf f} \in S_{d}({\mathbb{R}}),\, z \in {\mathbb{C}}$, $n \in {\mathbb{N}}$.
Then $(\Phi_n)_{n\in {\mathbb{N}}}$ converges strongly in $(S)'$ to a unique Hida distribution.
Let $\,{\bf{B}}(t)$, $t\geq 0$, be the $d$-dimensional Brownian motion as in . Consider $$\frac{{\bf{B}}(t+h,\boldsymbol{\omega}) - {\bf{B}}(t,\boldsymbol{\omega})}{h} = (\langle \frac{{\mathbf{1}}_{[t,t+h)}}{h} , \omega_1 \rangle , \dots (\langle \frac{{\mathbf{1}}_{[t,t+h)}}{h} , \omega_d \rangle),\quad h>0.$$ Then in the sense of Corollary \[seqcor\] it exists $$\begin{aligned}
\langle {\boldsymbol\delta_t}, {\boldsymbol \omega} \rangle := (\langle \delta_t,\omega_1 \rangle, \dots ,\langle \delta_t,\omega_d \rangle):= \lim_{h\searrow 0} \frac{{\bf{B}}(t+h,\boldsymbol{\omega}) - {\bf{B}}(t,\boldsymbol{\omega})}{h}.\end{aligned}$$ Of course for the left derivative we get the same limit. Hence it is natural to call the generalized process $\langle {\boldsymbol\delta_t}, {\boldsymbol \omega} \rangle$, $t\geq0$ in $(S)'$ vector valued white noise. One also uses the notation ${\boldsymbol \omega}(t) =\langle{\boldsymbol
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4(q)$, $|T_1|$ divides $q^6-1$), for all groups except ${}^2B_2(q), {}^2G_2(q), {}^2F_4(q) $. In the latter cases, denoted by $(\star)$, the information can be obtained from [@VV2 Lemma 2.8]
The previous lemmas provide the following result on the center of the prime graph of a simple group of Lie type, which can also be derived from [@center Proposition 2.9].
If $N$ is a simple group of Lie type, then ${{\operatorname}{\mathcal{Z}}(\Gamma(N))}=\emptyset$.
Let $t$ be the characteristic of the group of Lie type $N$. It is well known that $t \not \in {{\operatorname}{\mathcal{Z}}(\Gamma(N))}$. If $p \in {{\operatorname}{\mathcal{Z}}(\Gamma(N))}$, then for each prime $r \neq t$, there exists an abelian $t'$-subgroup of $N$ whose order is divisible by $p$ and $r$. Since any abelian $t'$-subgroup is contained in a maximal torus, this means that $p \in \pi(T)$ for each maximal torus $T$ of $N$. Hence, the information given in Lemmas \[class\] and \[excep\] leads to a contradiction.
The minimal counterexample: reduction to the almost simple case
===============================================================
In this section we will give a description of the structure of a minimal counterexample to our Main Theorem. Hence, having in mind Lemma \[pclos\], we assume the following hypotheses:
(H1)
: $p$ is a prime number.
(H2)
: $G$ is a group satisfying the following conditions:
1. $G=AB$ is the product of the subgroups $A$ and $B$, and $p$ does not divide $i_G(x) $ for every $p$-regular element of prime power order $x \in A \cup B$.
2. $G$ does not have a normal Sylow $p$-subgroup.
Among all such groups we choose $(G, A, B)$ such that $|G|+|A|+|B|$ is minimal.\
For such a group $G$ we have the following results.
\[0\] $G$ has a unique minimal normal subgroup $N$ which is not a $p$-group. Moreover, $P \neq 1$, $PN \unlhd G$, $G/N=PN/N \times O_{p'}(G/N)$, and $G=NN_{G}(P)$, for each $P \in {{\operatorname}{Syl}_{p}\left(G\right)}$.
Since the hypotheses (H2)(i) are clearly inherited by quotients
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You only need to annotate the type of an empty list. Lists with at least one
element don't require a type annotation because the type can be inferred from
the type of that element.
Dhall does not have buit-in support for homogeneous maps. Dhall does have
statically typed heterogeneous records (i.e. something like `{ foo = Bool, bar
= "ABC" }` which has type `{ foo : Bool, bar : Text }` for example).
If you want to store different type of values in the same list you wrap them
in a union. For example, if you want to store both `Text` values and `Natural`
numbers in a list you would do:
let union = constructors < Left : Natural | Right : Text >
in [ union.Left 10, union.Right "ABC", union.Right "DEF", union.Left 4 ]
The closest thing to a homogeneous map in Dhall is an association list of type
`[ { mapKey : Text, mapValue : a } ]` but even that is still not an exact fit
since it doesn't guarantee uniqueness of keys. However, Dhall's JSON/YAML
integration does convert that automatically to a JSON/YAML homogeneous map
(i.e. a JSON record where every field has the same type).
In general, Dhall's JSON/YAML integration has several tricks and conventions
that translate to weakly typed JSON idioms (such as homogeneous maps, omitting
null values, and using tags).
~~~
jarpineh
Thank you for your response.
This Left and Right declaration style was a new one for me. Also, homogeneous
map wasn't exactly a familiar concept. I don't remember meeting these when
learning TypeScript and dabbling with Elm. I fear I don't quite grasp the type
structure here yet. I can go forward with my testing based on your example.
~~~
KirinDave
By the way, it's entirely fair game for you to create config/domain specific
things and not simply (Left|Right) dichotomies.
And by the way, TypeScript DOES have a form of Sum typing like that as of
2017! You can say something like this from the manual:
type Shape = Square | Rectangle | Circle | Triangle;
function area(s: Shape) {
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eraction, i.e. they develop bulge rotation, bars and spiral arms in the first few hundred Myr of the simulation.
[^2]: We should note, however, that most $2\sigma$-galaxies do not exhibit a centrally peaked velocity dispersion, like the one presented here.
[^3]: The suggested mechanism could be responsible for the formation of KDCs in non-retrograde close encounters, e.g.[@Barnes_2002].
---
abstract: 'We study the hydrodynamics of relativistic fluids with several conserved global charges (i.e., several species of particles) by performing a Kaluza-Klein dimensional reduction of a neutral fluid on a $N$-torus. Via fluid/gravity correspondence, this allows us to describe the long-wavelength dynamics of black branes with several Kaluza-Klein charges. We obtain the equation of state and transport coefficients of the charged fluid directly from those of the higher-dimensional neutral fluid. We specialize these results for the fluids dual to Kaluza-Klein black branes.'
author:
- Adriana Di Dato
title: '**Kaluza-Klein reduction of relativistic fluids and their gravity duals**'
---
*Departament de Física Fonamental and\
Institut de Ciències del Cosmos, Universitat de Barcelona,\
Martí i Franquès 1, ES-08028, Barcelona, Spain.*\
Introduction {#1}
============
Kaluza-Klein dimensional reduction is a well known method to obtain solutions to a gravitational theory coupled to a Maxwell field, plus a scalar (dilaton) field. Velocities (or momenta) along the compactified direction result in electric charges in the reduced theory [@Pope]. Thus, if we take a neutral black string solution of the vacuum Einstein theory, perform a boost along the direction of the string and then dimensionally reduce in this direction, we obtain an electrically charged black hole of the Einstein-Maxwell-dilaton theory, for a particular value of the dilaton coupling [@Horowitz].
It should be clear that this method is not exclusive to gravitational theories. The identification between momenta along the internal direction and conserved char
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s used to estimate value of $\lambda$ by applying Hill’s method [@gopi.personal]. The choice $p=0.03$ provides results in line with Ref. [@gopi.volume], for $\Delta t = 15$ min time windows one finds $\lambda = 1.67 \pm 0.20$. There are several issues with this approach:
1. $p$ is a parameter that can be chosen arbitrarily. With the variation of $p$ the same procedure can produce estimates ranging from $\lambda = 1.1\pm 0.2$ ($p=0.10$) to $\lambda = 2.15 \pm 0.2$ ($p=0.005$).
2. The transformation significantly decreases the estimates of $\lambda$, down to the range of Levy stable distributions ($\lambda < 2$). Estimates for the untransformed data are given in Table \[tab:DETRlambda94-95\] for comparison.
It is simple to show, that the first issue emerges, i.e. the estimates systematically depend on $p$, when one applies Hill’s method to a finite sample from a distribution of the form $${\mathbb P}_{\Delta t}(f) \propto (f+f_0)^{-(\lambda + 1)},
\label{eq:pl2}$$ where $f_0$ is a non-zero constant. The transformation to $f_i(t)-\ev{f_i}$ does not resolve the problem, but biases the estimates further.
Instead, to correct for these biases one can (i) either find the proper constant $f_0$, remove it from the data, and apply Hill’s estimator afterwards (ii) or apply the estimator of Fraga-Alves [@alves], which is insensitive to such shifts. Both of these estimates were found to be significantly higher [^1]: $\lambda > 2$, see Table \[tab:DETRlambda94-95\]. The methods are described in detail in Ref. [@eisler.sizematters].
The two corrected estimators show a strong tendency of increasing $\lambda$ with increasing $\Delta t$. Monte Carlo simulations on surrogate datasets show that this is beyond what could be explained by decreasing sample size. For distributions with $\lambda < 2$ increasing window size should result in a convergence to the corresponding Levy distribution, and the measured $\lambda$’s should be independent of $\Delta t$. Only when $\lambda > 2$ can the measured effective value of $\lambd
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der. Pick $g_1,g_2,g_3\in G$ and let us check that $D(g_1H,g_3H)\leq D(g_1H,g_2H)+D(g_2H,g_3H)$. Choose an arbitrary $\varepsilon>0$ and and some $h_1,h_2, h'_2,h_3\in H$ such that $D(g_1H,g_2H)\geq d(g_1h_1,g_2h_2)-\varepsilon$ and $D(g_2H,g_3H)\geq d(g_2h'_2,g_3h_3)-\varepsilon$. Then $$D(g_1H,g_3H)\leq d(g_1h_1,g_3h_3(h'_2)^{-1}h_2)\leq d(g_1h_1,g_2h_2)+d(g_2h_2,g_3h_3(h'_2)^{-1}h_2)=$$ $$d(g_1h_1,g_2h_2)+d(g_2h'_2,g_3h_3)\leq D(g_1H,g_2H)+D(g_2H,g_3H)-2\varepsilon.$$ Since $\varepsilon$ was arbitrary, we are done.
We note that when $G$ is a metrizable group and $H$ is a compact subgroup, then a compatible left-invariant and right $H$-invariant metric on $G$ always exists. Indeed, let $d$ be an arbitrary compatible left-invariant metric on $G$. We define, for $g,f\in G$, $D(g,f):=\max_{h\in H} d(gh,fh)$. Alternatively, using a normalized invariant Haar measure $\mu$ on $H$, we can define $D$ by averaging as follows: $D(g,f):=\int_H d(gh,fh)d\mu(h)$. We leave to the reader to check that both formulas define a compatible left-invariant and right $H$-invariant metrics.
Our main tool in this subsection will be the following proposition. We note that simultaneously while writing this paper, the content of the proposition is being developed into a more general form in [@AACD].
\[prop:projection\] Let $G$ be a topological group equipped with a compatible metric $d$ and a compact subgroup $H$. Suppose, additionally, that at least one of the following conditions holds:
(i) $d$ is left-invariant and $H$ is normal,
(ii) $d$ is right-invariant and $H$ is normal, or
(iii) $d$ is left-invariant and right $H$-invariant.
Then there exists a norm one projection $P:{\mathcal{F}}(G,d)\rightarrow {\mathcal{F}}(G,d)$ ranging onto a linearly isometric copy of ${\mathcal{F}}(G/H,D)$, where $D$ is the quotient metric as defined in .
By the discussion preceding the statement of the proposition, it is verified that $D$ is a well-defined metric. Let $\mu$ be the normalized invariant Haar measure on $H$. If (ii) or (iii) holds, w
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s statement implies Theorem \[main\].
The first part of Theorem \[mainmain\] has been established above. In order to prove the second part, we will define a simple notion of ‘equivalence’ of germs (Definition \[equivgermsnew\]), such that, in particular, equivalent germs $\alpha(t)$ lead to the same component of the PNC. We will show that any given germ $\alpha(t)$ centered at a point of ${{\mathscr S}}$ either is equivalent (after a parameter change, if necessary) to one of the marker germs, or its lift in ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$ meets the PNC at a point of $R$ (cf. Remark \[eluding\]) or of the boundary of the orbit of a marker center. In the latter cases, the center of the lift varies in a locus of dimension $<7$, hence such germs do not contribute components to the PNC. The following lemma allows us to identify easily limits in the intersection of $R$ and the PNC.
\[rank2lemma\] Assume that $\alpha(0)$ has rank $1$. If $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ is a star with center on $\ker\alpha(0)$, then it is a rank-2 limit.
Assume ${{\mathscr X}}=\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ is a star with center on $\ker\alpha(0)$. We may choose coordinates so that $x=0$ is the kernel line, and the generator for the ideal of ${{\mathscr X}}$ is a polynomial in $x,y$ only. If $$\alpha(t)=\begin{pmatrix}
a_{11}(t) & a_{12}(t) & a_{13}(t) \\
a_{21}(t) & a_{22}(t) & a_{23}(t) \\
a_{31}(t) & a_{32}(t) & a_{33}(t)
\end{pmatrix}\quad,$$ then ${{\mathscr X}}=\lim_{t\to 0}{{\mathscr C}}\circ\beta(t)$ for $$\beta(t)=\begin{pmatrix}
a_{11}(t) & a_{12}(t) & 0 \\
a_{21}(t) & a_{22}(t) & 0 \\
a_{31}(t) & a_{32}(t) & 0
\end{pmatrix}\quad.$$ Since $\alpha(0)$ has rank 1 and kernel line $x=0$, $$\alpha(0)=\begin{pmatrix}
a_{11}(0) & 0 & 0 \\
a_{21}(0) & 0 & 0 \\
a_{31}(0) & 0 & 0
\end{pmatrix}=\beta(0)\quad.$$ Now $\beta(t)$ is contained in the rank-2 locus, verifying the assertion.
A limit $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ as in this lemma will be called a ‘kernel star’.
Sec
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to their results.
This paper is organized as follows. In Section 1 we will prove Theorem \[main\] and derive Corollaries \[bgln\] and \[string\]. In Section 2 we describe the $K$-theoretic implications of Thoerem \[main\], and in particular we prove Theorem \[ktheory\].
Automorphisms of $R$-module bundles
===================================
Let $R$ be a ring spectrum. Let ${\mathcal{E}}\to X$ be an $R$-module bundle of rank $n$, in the sense of Lind [@lind]. This is a parameterized spectrum over $X$ in the sense of May and Sigurdsson [@maysigurd], where for each $x \in X$, the fiber ${\mathcal{E}}_x$ is an $R$-module spectrum of rank $n$. We denote the category of such bundles by $R-mod_n (X)$. Again, this category was defined in [@lind]. It was shown there that equivalence classes of such bundles are classified by homotopy classes of maps $X \to BGL_n(R)$. Fix a particular map $\gamma_{\mathcal{E}}: X \to BGL_n(R)$ classifying ${\mathcal{E}}$. This choice defines a basepoint in the mapping space $\gamma_{\mathcal{E}}\in Map_{\mathcal{E}}(X, BGL_n(R))$. The endomorphisms of ${\mathcal{E}}$ in $R-mod_n(X)$ is a parameterized spectrum which we denote by $End_M^R({\mathcal{E}}) \to X$. For every $x \in X$ this defines a fiber spectrum $End_M^R({\mathcal{E}})_x$ which is equivalent to the ring of endomorphisms $End^R(\vee_n R)$. $End_M^R({\mathcal{E}}) $ is a parameterized ring spectrum under composition. By taking a fibrant replacement if necessary, we can take sections to produce an ordinary spectrum $$End^R({\mathcal{E}}) = \Gamma_M(End^R_M ({\mathcal{E}})).$$ The parameterized ring structure on $End_M^R({\mathcal{E}})$ defines a ring spectrum structure on $End^R({\mathcal{E}})$.
\[haut\] We define the group-like monoid ${hAut^R}({\mathcal{E}})$ to be the units of the ring spectrum of endomorphisms, $${hAut^R}({\mathcal{E}}) = GL_1(End^R({\mathcal{E}})).$$
We are now ready to prove Theorem \[main\].
Consider the fiber bundle of infinite loop spaces given by taking the zero spaces of the fibrant model of $End
| 2,824
| 1,368
| 2,637
| 2,786
| 3,484
| 0.772024
|
github_plus_top10pct_by_avg
|
values. (**A**) Hazelnut tested in 1 µL of sample in 99 µL of running buffer (RB) (**B**) Peanut tested in 1 µL of sample in 99 µL of RB. (**C**) Hazelnut tested in 25 µL sample in 75 µL of RB. (**D**) Peanut tested in 25 µL sample in 75 µL of RB. (**E**) Hazelnut tested in 75 µL sample in 25 µL of RB. (**F**) Peanut tested in 75 µL of sample in 25 µL of RB. Error bars show standard deviation (SD) from triplicate measurements.](biosensors-09-00143-g004){#biosensors-09-00143-f004}
{#biosensors-09-00143-f005}
biosensors-09-00143-t001_Table 1
######
Comparison of optimized Flow-through and Lateral Flow parameters (RB \*).
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Parameter Passive Flow-through Active Flow-through PHC \*\* HPC \*\*
------------------------------------------------------------- ---------------------------------------------- ------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------- -------
| 2,825
| 147
| 3,058
| 3,199
| null | null |
github_plus_top10pct_by_avg
|
`VD+LDVYSDAY`
`VDLLDVYSDAY`
`SRR022865_54601` `3` `ggttggttggt`
1180638 NONSYN T:5 G:97 G:37 `tattatacaca` cell division protein ftsA
`ttaattcttat`
`gi 87160920 ref` `104` `KGDIIGYVEAMK`
`KGDIIGYVEA+K`
`KGDIIGYVEAIK`
`SRR022865_82913` `-37` `aggaagtgggaa` acetyl-CoA carboxylase, biotin carboxyl carrier protein
`agattgatacta`
1714319 NONSYN T:6 C:95 C:37 `gattaattagaa`
| 2,826
| 6,953
| 552
| 715
| null | null |
github_plus_top10pct_by_avg
|
gle\left [\xi_{\rm eq}(t)
+g'(x)\xi_{\rm neq}(t)\right ]\left [\frac{\partial}{\partial v^{-\tau}}\{
\xi_{\rm eq}(t-\tau)+g'(x^{-\tau})\xi_{\rm neq}(t-\tau)\}\right]\rangle
p \hspace{0.2cm},\end{aligned}$$ where we have used the fact that the Jacobian obey the equation $^{12}$ $$\frac{d}{dt}\log\left|\frac{d(x^{t},v^{t})}{d(x,v)}\right|
=\frac{\partial}{\partial x}v+\frac{\partial}{\partial v}\{-\Gamma v+\tilde{V}
'(x)\} =-\Gamma$$ so that Jacobian equals to $e^{-\Gamma t}$.
As a next approximation we consider the ‘unpurterbed’ part of Eq.(16) and take the variation of $v$ during $\tau_{c}$ into account to first order in $\tau_{c}$. Thus we have $$x^{-\tau}=x-\tau v\hspace{0.2cm};\hspace{0.2cm}v^{-\tau}=v+\Gamma\tau v+\tau
\tilde{V}'(x)\hspace{0.2cm}.$$
Neglecting terms ${\cal O}(\tau^{2})$ Eq.(32) yields, $$\frac{\partial}{\partial v^{-\tau}}=(1-\Gamma\tau)\frac{\partial}{\partial v}
+\tau \frac{\partial}{\partial x}\hspace{0.2cm}.$$
Taking into consideration of Eq.(33), Eq.(30) can be simplified after some algebra to the following form, $$\begin{aligned}
\frac{\partial}{\partial t}p(x,v,t)=-\frac{\partial}{\partial x}(vp)+
\frac{\partial}{\partial v}\left\{ \Gamma (x)v+\tilde{V}'(x)-2g'(x)g''(x)
I_{nn}\right \}p\nonumber\\
\nonumber\\
+\left \{ I_{ee}+[g'(x)]^{2}I_{nn} \right \}\frac{\partial^{2} p}{\partial
v\partial x}\hspace{5.0cm}\nonumber\\
\nonumber\\
+\left \{ J_{ee}-\Gamma(x)I_{ee}+[g'(x)]^{2}J_{nn}-\Gamma(x)[g'(x)]^{2}I_{nn}
-vg'(x)g''(x)I_{nn}\right\}\frac{\partial^{2} p}{\partial v^{2}}\hspace{0.2cm},\end{aligned}$$ where, $$\left.\begin{array}{l}
I_{ee}=\int_{0}^{\infty} d\tau \langle\xi_{\rm eq}(t)\xi_{\rm eq}(t-\tau)\rangle\tau \\
I_{nn}=\int_{0}^{\infty} d\tau \langle\xi_{\rm neq}(t)\xi_{\rm neq}(t-\tau)\rangle\tau \\
J_{ee}=\int_{0}^{\infty} d\tau \langle\xi_{\rm eq}(t)\xi_{\rm eq}(t-\tau)\rangle \\
J_{nn}=\int_{0}^{\infty} d\tau \langle\xi_{\rm neq}(t)\xi_{\rm neq}(t-\tau)\rangle \\
\end{array}\right\}\hspace{0.2cm}.$$
The subscripts $ee$ and $nn$ in the above expressions for the i
| 2,827
| 2,666
| 2,695
| 2,852
| null | null |
github_plus_top10pct_by_avg
|
} A W \right\}_{k K}
\left\{ W ^{\dagger} A (UX) \right\}_{K m}
\left\{ (UX)^{\dagger} A W \right\}_{m L}
\left\{ W ^{\dagger} A (UX) \right\}_{L l}
\biggr\}.
\label{P-beta-alpha-W4-H4-double}\end{aligned}$$
$$\begin{aligned}
&& P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{5th-1st}
\equiv
2 \mbox{Re} \left[ \left(
S^{(0)}_{\alpha \beta} \right)^{*}
S_{\alpha \beta}^{(4)} [3]_\text{First}
\right]
\nonumber \\
&=&
2 \mbox{Re}
\biggl\{
- \sum_{n}
\sum_{k L}
\frac{ 1 }{ \Delta_{L} - h_{k} }
\left[ (ix) e^{- i ( \Delta_{L} - h_{n} ) x} + \frac{ e^{- i ( \Delta_{L} - h_{n} ) x} -
e^{- i ( h_{k} - h_{n} ) x} }{ ( \Delta_{K} - h_{k} ) }
\right]
\nonumber \\
&\times&
(UX)_{\alpha k} W^*_{\beta L}
(UX)^*_{\alpha n} (UX)_{\beta n}
\left\{ (UX)^{\dagger} A W \right\}_{k L}
\left\{ W^{\dagger} A W \right\}_{L L}
\nonumber \\
&+&
\sum_{n}
\sum_{k L}
\sum_{K \neq L}
\frac{ 1 }{ ( \Delta_{L} - \Delta_{K} ) ( \Delta_{L} - h_{k} ) ( \Delta_{K} - h_{k} ) }
\nonumber \\
&\times&
\biggl[
\left( \Delta_{K} - h_{k} \right)
e^{- i ( \Delta_{L} - h_{n} ) x}
- \left( \Delta_{L} - h_{k} \right)
e^{- i ( \Delta_{K} - h_{n} ) x}
- \left( \Delta_{K} - \Delta_{L} \right)
e^{- i ( h_{k} - h_{n} ) x}
\biggr]
\nonumber \\
&\times&
(UX)_{\alpha k} W^*_{\beta L}
(UX)^*_{\alpha n} (UX)_{\beta n}
\left\{ (UX)^{\dagger} A W \right\}_{k K}
\left\{ W^{\dagger} A W \right\}_{K L}
\nonumber \\
&-&
\sum_{n}
\sum_{k L}
\frac{ 1 }{ ( \Delta_{L} - h_{k} )^2 }
\biggl[
(ix) \left( e^{- i ( h_{k} - h_{n} ) x} + e^{- i ( \Delta_{L} - h_{n} ) x} \right)
+ 2 \frac{e^{- i ( \Delta_{L} - h_{n} ) x} - e^{- i ( h_{k} - h_{n} ) x} }{ ( \Delta_{L} - h_{k} ) }
\biggr]
\nonumber \\
&\times&
(UX)_{\alpha k} W^*_{\beta L}
(UX)^*_{\alpha n} (UX)_{\beta n}
\left\{ (UX)^{\dagger} A W \right\}_{k L}
\left\{ W^{\dagger} A (UX) \right\}_{L k}
\left\{ (UX)^{\dagger} A W \right\}_{k L}
\nonumber \\
&+&
\sum_{n}
\sum_{k L}
\sum_{m \neq k}
\biggl[
- \frac{ (ix) e^{- i ( \Delta_{L} - h_{n} ) x} }{ ( \Delta_{L} - h_{k} )( \Delta_{L} -
| 2,828
| 2,646
| 2,551
| 2,805
| null | null |
github_plus_top10pct_by_avg
|
}\cdot {\ensuremath{\operatorname{\mathbf{Pr}}\left[i\in D_t\right]}}.
\end{aligned}$$ In order to have $i\in D_t$, first an edge containing $i$ must be selected, and then the chosen $d$-element subset of that edge must contain $i$. By the $\beta$-balancedness property, $${\ensuremath{\operatorname{\mathbf{Pr}}\left[i\in D_t\right]}}{\leqslant}{\frac{\beta {s}}{n}}\cdot \frac{\binom{{s}-1}{d-1}}{\binom{{s}}{d}}{\leqslant}{ \frac{\beta }{n}}.$$ Using the above inequality, we simplify Inequality (\[ineq:first\]) as follows: $${\ensuremath{\operatorname{\mathbf{Pr}}\left[A(t,i)\right]}}{\leqslant}{\frac{6\beta}{n}} + {\frac{\beta d}{n}}\, {\ensuremath{\operatorname{\mathbf{Pr}}\left[{\mathbb{I}}_t=0 \mid i\in D_t \right]}}.$$ [If $d{\leqslant}6$ then the above inequality immediately implies that [${\ensuremath{\operatorname{\mathbf{Pr}}\left[A(t,i)\right]}}{\leqslant}{12\beta}/{n}$]{}. This completes the proof when $d{\leqslant}6$. ]{} For the remainder of the proof we assume that $d{\geqslant}7$, and prove that $$\begin{aligned}
\label{u:cl}
{\ensuremath{\operatorname{\mathbf{Pr}}\left[{\mathbb{I}}_t=0 \mid i\in D_t \right]}} {\leqslant}\hat{c}/d
\end{aligned}$$ for some absolute constant $\hat{c}>0$. [From this, we see that ${\ensuremath{\operatorname{\mathbf{Pr}}\left[A(t,i)\right]}} {\leqslant}\alpha/n$ where [$\alpha = \beta(6 + \hat{c})$]{}. As $i$ was an arbitrary bin, this proves that the process is $(\alpha,m)$-uniform.]{}
Let ${\mathcal{F}}$ be the event that $H_t$ contains at least ${s}/2$ empty vertices for all $t=1,\ldots, m$. By Lemma \[lem:empty\], we have ${\ensuremath{\operatorname{\mathbf{Pr}}\left[{\mathcal{F}}\right]}}{\geqslant}1-n^{-2}$. Then $$\begin{aligned}
\label{ineq:sec}
&{\ensuremath{\operatorname{\mathbf{Pr}}\left[{\mathbb{I}}_t=0 \mid i\in D_t\right]}} \nonumber \\&={\ensuremath{\operatorname{\mathbf{Pr}}\left[{\mathbb{I}}_t=0 \mid (i\in D_t) \text{~and~} {\mathcal{F}}\right]}}\cdot{\ensuremath{\operatorname{\mathbf{Pr}}\left[{\mathcal{F}}\right]}}+{\ensuremat
| 2,829
| 1,363
| 2,330
| 2,600
| null | null |
github_plus_top10pct_by_avg
|
------------------------------------------------------------------------
Figure \[fig::confint\] shows typical confidence intervals for the projection parameter, $\beta_{{\widehat{S}}}$, and the LOCO parameter, $\gamma_{{\widehat{S}}}$, for one realization of each Setting. Notice that confidence intervals are only constructed for $j\in {\widehat{S}}$. The non-linear term is successfully covered in Setting B, even though the linear model is wrong.
![*Joint coverage probability of the intervals for $\beta_{{\widehat{S}}}$ and $\gamma_{{\widehat{S}}}$ in Setting B, as sample size $n$ varies with $p=50$ held fixed. The coverage for $\gamma_{{\widehat{S}}}$ is accurate even at low sample sizes, while the coverage for $\beta_{{\widehat{S}}}$ converges more slowly.* []{data-label="fig::coverage"}](coverage_plot.pdf)
Figure \[fig::coverage\] shows the coverage probability for Setting B as a function of $n$, holding $p=50$ fixed. The coverage for the LOCO parameter, $\gamma_{{\widehat{S}}}$ is accurate even at low sample sizes. The coverage for $\beta_{{\widehat{S}}}$ is low (0.8-0.9) for small sample sizes, but converges to the correct coverage as the sample size grows. This suggests that $\beta_{{\widehat{S}}}$ is an easier parameter to estimate and conduct inference on.
Berry-Esseen Bounds for Nonlinear Parameters With Increasing Dimension {#section::berry}
======================================================================
The results in this paper depend on a Berry-Esseen bound for regression with possibly increasing dimension. In this section, there is no model selection or splitting. We set $d=k$ and $S = \{1,\ldots, k\}$ where $k < n$ and $k$ can increase with $n$. Later, these results will be applied after model selection and sample splitting. Existing Berry-Esseen results for nonlinear parameters are given in [@pinelis2009berry; @shao2016stein; @chen2007normal; @anastasiou2014bounds; @anastasiou2015new; @anastasiou2016multivariate]. Our results are in the same spirit but we keep careful track of the effe
| 2,830
| 1,489
| 1,067
| 2,508
| 989
| 0.796219
|
github_plus_top10pct_by_avg
|
imply because the associated graded ring of $U_c$ is ${\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]^{{W}}$. The results we need follow easily from the corresponding results on $\operatorname{Hilb^n{\mathbb{C}}^2}$ and so we begin with the latter.
The (isospectral) Hilbert scheme {#isospecsec}
--------------------------------
Following [@hai3 Definition 3.2.4] [*the isospectral Hilbert scheme*]{} ${\mathbb{X}}_n$\[isospec-defn\] is the reduced fibre product $$\begin{CD} {\mathbb{X}}_n @> f_1 >> {\mathbb{C}}^{2n}
\\ @V \rho_1 VV @VVV \\ \operatorname{Hilb^n{\mathbb{C}}^2}@> \tau >> {\mathbb{C}}^{2n}/{{W}}. \end{CD}$$ It is a highly non-trivial fact (see [@hai3 Theorem 3.1 and the proof of Proposition 3.7.4]) that $\rho_1$ is a flat map of degree $n!$.
Haiman has given a description of both $\operatorname{Hilb^n{\mathbb{C}}^2}$ and ${\mathbb{X}}_n$ as Proj of appropriate graded rings and we recall this description since it will be extremely important to us. Let $ {\mathbb{A}}^1={\mathbb{C}}[{\mathbb{C}}^{2n}]^{{\epsilon}}$ \[AAA-1-defn\] be the space of ${{W}}$-alternating polynomials in ${\mathbb{C}}[{\mathbb{C}}^{2n}]$ and write $ {\mathbb{J}}^1= {\mathbb{C}}[{\mathbb{C}}^{2n}]{\mathbb{A}}^1$ for the ideal generated by ${\mathbb{A}}^1$. For $d\geq 1$ define ${\mathbb{A}}^d$ and ${\mathbb{J}}^d$ to be the respective $d^{\text{th}}$ powers of ${\mathbb{A}}^1$ and ${\mathbb{J}}^1$ using multiplication in ${\mathbb{C}}[{\mathbb{C}}^{2n}]$; thus \[JJJ-defn\] ${\mathbb{J}}^d={\mathbb{C}}[{\mathbb{C}}^{2n}] {\mathbb{A}}^d$. Finally, set ${\mathbb{J}}^0={\mathbb{C}}[{\mathbb{C}}^{2n}]$, ${\mathbb{A}}^0={\mathbb{C}}[{\mathbb{C}}^{2n}]^{{W}}$ and ${\mathbb{A}}=\bigoplus_{d\geq 0} {\mathbb{A}}^d\cong {\mathbb{A}}^0[t{\mathbb{A}}^1]$. Then [@haidis Proposition 2.6] proves that $$\label{AAA-alg-defn}
\operatorname{Hilb^n{\mathbb{C}}^2}\cong \operatorname{Proj}{\mathbb{A}}\mathrm{ \ as\ a\ scheme\ over\ }
\operatorname{Spec}{\mathbb{A}}^0 =
{\mathbb{C}}^{2n}/{{W}}$$ Similarly, ${\mathbb{X}}_n \cong \operatorname{
| 2,831
| 2,058
| 1,550
| 2,698
| null | null |
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|
{h}}{ \,{}_{^{^{\bullet}}}}b_2)$ and $[{\mathbf{E}}, b_1b_2] = [{\mathbf{E}},b_1]b_2 + b_1[{\mathbf{E}},b_2]$. By induction, it therefore suffices to prove the result when $b=em\delta e\in B_{k,k-1}=
eH_{c+k} \delta e$, for some $k>0$. By we see that ${\mathbf{h}}{ \,{}_{^{^{\bullet}}}}b = e[{\mathbf{h}}_{c+k}, m]\delta e$ whereas $[{\mathbf{E}}, b ] = e[{\mathbf{E}}, m] \delta e + em[{\mathbf{E}}, \delta] e$. By , $ [{\mathbf{h}}_{c+k}, m]=[{\mathbf{E}}, m]$ and so the two gradings differ by $\operatorname{{\mathbf{E}}\text{-deg}}\delta = N$.
\(2) This follows from part (1) combined with Proposition \[poincare-SA\], respectively .
{#filter-injA}
Fix $k\geq 0$ and for notational simplicity write $\mathcal{J}=e J^k\delta^k$ and $\mathcal{N}=N(k)$. The final step in the proof of Proposition \[pre-cohh\] is to show that the inclusion $\Theta: \mathcal{J} \hookrightarrow \operatorname{{\textsf}{ogr}}\mathcal{N}$ from Lemma \[thetainjA\](3) is surjective. In order to effectively use Corollary \[poincare-S2A\], we do this by lifting $\Theta$ to a ${\mathbb{C}}[{\mathfrak{h}}]^{{{W}}}$-module map $\theta: \mathcal{J}\to \mathcal{N}$.
The order filtration on $D({\mathfrak{h}^{\text{reg}}})\ast {{W}}$ induces a graded structure on $\operatorname{{\textsf}{ogr}}D({\mathfrak{h}^{\text{reg}}})\ast {{W}}\cong
{\mathbb{C}}[{\mathfrak{h}^{\text{reg}}}\oplus{\mathfrak{h}}^*]\ast{{W}}$ and hence on $\operatorname{{\textsf}{ogr}}{\mathcal{N}}$, which we call the *order gradation*; thus $\deg_{\operatorname{{\textsf}{ord}}} ({\mathbb{C}}[{\mathfrak{h}}]\ast {{W}})=0$, while $\deg_{\operatorname{{\textsf}{ord}}} {\mathfrak{h}}=1$. We will use the same terminology for the induced grading on the rings $A^0={\mathbb{C}}[{\mathfrak{h}}\oplus {\mathfrak{h}}^*]^W$ and $A$ and the module ${\mathcal{J}}$.
Let ${\mathcal{N}}^m=\operatorname{{\textsf}{ord}}^m{\mathcal{N}}$ denote the elements in ${\mathcal{N}}$ of order $\leq m$. Similarly, write ${\mathcal{J}}= \bigoplus_{m\geq 0} \operatorname{{\textsf}{ogr}}^m {\mathcal{J}}$ fo
| 2,832
| 1,853
| 1,500
| 2,854
| null | null |
github_plus_top10pct_by_avg
|
_k+e_l+\Gamma_i, & \mbox{if} & k\neq l, \end{array} \right.$$ Thus $$\Gamma(a_{F_i})\setminus\Gamma(a_{F_1},\ldots,a_{F_{i-1}})=\Union_{L\in\mathcal L}(e_L+\Gamma_i),$$ where $${\mathcal L}=\{\{k_1,\ldots, k_{i-1}\}\: k_j\in F_i\setminus F_j \text{ for } j=1,\ldots,{i-1}\}$$ and where $e_L=\sum_{j\in L}e_{j}$ for each $L\in\mathcal L$.
The union $$\Union_{L\in\mathcal L}(e_L+\Gamma_i)$$ is a Stanley set if and only if there exists $L\in\mathcal L$ such that $e_{L'}+\Gamma_i\subset e_L+\Gamma_i$ for all $L'\in \mathcal L$, and this is the case if and only if there exists $L\in\mathcal L$ such that $L\subset L'$ for all $L'\in \mathcal L$.
We claim that the last condition is equivalent to the condition that all facets of $\langle F_i\rangle\sect \langle F_1,\ldots, F_{i-1}\rangle$ are maximal proper subfaces of $\langle F_i\rangle$.
Suppose first that there is a set $L_0\in\mathcal L$ which is minimal under inclusion. We may assume that $L_0=[m]$. Let $k\in [m]$ and assume that all sets $F_i\setminus F_j$ which contain $k$ have more than one element. Then for each such set we can pick $k_j\in F_i\setminus F_j$ with $k_j\neq k$, and hence there exists $L\in\mathcal L$ which does not contain $k$, a contradiction, since $k\in L_0\subset L$. Thus for each $k\in L_0$ there exists an integer $j_k\in
[i-1]$ such that $F_i\setminus F_{j_k}=\{k\}$. Now let $j\in [i-1]$ be arbitrary. If $|F_i\setminus F_j|=1$, then by definition of the sets $L$, the set $F_i\setminus F_j$ is a subset of each $L$, and in particular of $L_0$. Thus we see that the subfaces of $\langle F_i\rangle\sect \langle F_1,\ldots,
F_{i-1}\rangle$ of codimension $1$ are exactly the faces $F_i\setminus\{k\}$ for $k=1,\ldots, m$. Suppose now there exists $j\in [i-1]$ for which $F_i\sect
F_j$ is not contained in any of these codimension 1 subfaces of $F_i$ (in which case not all facets of $\langle F_i\rangle\sect \langle F_1,\ldots,
F_{i-1}\rangle$ would be maximal proper subfaces of $\langle F_i\rangle$.). Then $k\not\in F_i\setminus F_j$ for $k=1,\ldots,
| 2,833
| 2,154
| 2,448
| 2,579
| 4,024
| 0.768584
|
github_plus_top10pct_by_avg
|
0.733
WBC (10^9^/mL) 6.2 ± 5.7 5.7 ± 1.4 0.560 4.9 ± 1.2 6.0 ± 3.3 0.117
PLT (10^9^/mL) 170.2 ± 58.4 197.5 ± 63.2 0.016 150.3 ± 51.1 197.4 ± 62.3 0.001
ALT (U/L) 29.6 ± 8.8 24.6 ± 8.3 0.001 29.7 ± 8.0 25.3 ± 8.6 0.116
AST (U/L) 27.5 ± 6.0 23.6 ± 6.0 0.004 27.8 ± 6.4 24.1 ± 6.6 0.012
ALP (U/L) 70.4 ± 25.0 67.2 ± 17.9 0.463 78.7 ± 29.4 66.1 ± 17.3 0.004
GGT (U/L) 25.8 ± 19.4 20.3 ± 12.1 0.423 33.9 ± 25.4 19.6 ± 10.3 0.000
TB (g/L) 14.1 ± 6.9 13.4 ± 6.7 0.538 16.0 ± 7.2 13.2 ± 6.6 0.057
PT (s) 11.6 ± 1.0 11.3 ± 1.2 0.100 11.7 ± 1.0 11.3 ± 1.2 0.152
TT (s) 19.4 ± 1.7 19.0 ± 1.6 0.235 19.2 ± 1.8 19.1 ± 1.6
| 2,834
| 5,285
| 1,976
| 2,182
| null | null |
github_plus_top10pct_by_avg
|
nu}\Box R
\quad\; , \quad
\mathcal{M}_3=R^{\mu\nu}\Box^2R_{\mu\nu}
\\~
\\
\mathcal{M}_5=\nabla^{\mu}\Box R \nabla_{\mu} R
\;\;\, , \;\;
\mathcal{M}_6=\nabla_{\mu}\nabla_{\nu}\nabla_{\alpha}R \nabla^{\mu} R^{\nu\alpha}
\;\;\, , \;\;
\mathcal{M}_{10}=\big(\Box R \big)^2
\;\; \, , \;\;
\mathcal{M}_{11}=\nabla_{\mu}\nabla_{\nu}R \nabla^{\mu}\nabla^{\nu}R
\\~
\\
\mathcal{M}_{12}=\nabla^{\mu}\nabla^{\nu}R \Box R_{\mu\nu}
\quad\; , \quad
\mathcal{M}_{14}=\nabla_{\mu}\nabla_{\nu} R_{\alpha\beta} \nabla^{\mu}\nabla^{\nu} R^{\alpha\beta}
\quad\; , \quad
\mathcal{M}_{18}=R \, \curv{L}_1
\quad\; , \quad
\mathcal{M}_{19}=R \, \curv{L}_4
\\~
\\ \mathcal{M}_{20}= S \Box R
\quad\; , \quad
\mathcal{M}_{33}= R \, \curv{L}_8
\end{array}
\right.
$\
\
\
We also introduce the definitions $\mathcal{K}_{9}=R^{\mu\nu}R^{\alpha}_{\;\,\mu}R_{\nu}^{\;\,\beta\sigma\rho} R_{\rho\sigma\beta\alpha}$ $\,$, $\,$ $\mathcal{M}_{13}=\Box R_{\mu\nu} \Box R^{\mu\nu}$ $\,$ and $\,$ $\mathcal{M}_{16}=\nabla_{\mu}\nabla_{\nu} R_{\alpha\beta} \nabla^{\beta}\nabla^{\alpha} R^{\nu\mu}$ that will be usefull later for static spherically symmetric space-times.
### Linear Combination. $H^8$ correction.
Consider the sum of all independent order 8 scalars for FLRW space-time : $$\begin{aligned}
J=&\sum v_i \mathcal{K}_i + \sum x_j \mathcal{M}_j \end{aligned}$$ Here, we follow exactly what we did for order 6 scalars. We derive the equation of motion associated with the previous sum, and see what conditions on $(v_i , x_j)$ cancel the equation, such that we find the 10 equivalence relations that exist between the scalars of the reduced basis. Therefore, we can consider only the following independent scalars with respect to the equation of motion, $\big( \mathcal{K}_{1},\mathcal{K}_{10},\mathcal{K}_{11},\mathcal{K}_{12},\mathcal{M}_{1},\mathcal{M}_{11},\mathcal{M}_{12} \big)$ and the reduced sum : $$\begin{aligned}
J=&\sum\limits_{i=1, 10, 11, 12} v_i \mathcal{K}_i + \sum\limits_{j=1, 11,
| 2,835
| 2,831
| 2,370
| 2,549
| null | null |
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|
gcd}(g_k^*, x^{m_i}-1)=1$.
\[\]
(v) ${\mathcal R}_i=\bigoplus_{k=1}^s{\mathcal R}_{ij}$.
\[\]
(vi) For each $k=1,2,\ldots,s$, the mapping $\phi_{ik}:~{\mathcal R}_{ik}\rightarrow R/(g_k^{*d_{ik}})$, defined by $$\phi_{ik}:~fb_k\widehat{g}_k^*+(x^{m_i}-1)\mapsto f+(g_k^{*d_{ik}}), ~\mbox{where}\; f\in R,$$ is a well defined isomorphism of rings.
\[\]
(vii) ${\mathcal R}_i=R/(x^{m_i}-1)\cong \bigoplus_{j=1}^sR/(g_j^{*d_{ij}})$.
From (vi), we have a well defined $R$-module isomorphism $\Phi_k$ from $b_k\widehat{g}_k^*{\mathcal R}$ onto $R/(g_k^{*d_{ik}})\times \cdots \times R/(g_k^{*d_{ik}})$, which defined by $$\Phi_k:~(\alpha_1, \ldots, \alpha_l)\mapsto (\phi_{1k}(\alpha_1),\ldots,\phi_{lk}(\alpha_l)),~ \mbox{where}\; \alpha_i\in {\mathcal R}_{ik}, i=1,2,\ldots,l.$$ $\Phi_k$ can introduce a natural $R$-module isomorphism $\mu_k$ from $b_k\widehat{g}_k^*{\mathcal R}$ onto ${\mathcal M}_k$.
For any $c=(c_0,c_1,\ldots,c_l)\in {\mathcal R}$, from (v) we deduce $c=(b_1\widehat{g}_1^*c_1+\cdots+b_s\widehat{g}_s^*c_1, \ldots, \\ b_1\widehat{g}_1^*c_l+\cdots+b_s\widehat{g}_s^*c_l)=b_1\widehat{g}_1^*c+\cdots+b_s\widehat{g}^*_sc$, where $b_k\widehat{g}_k^*c\in b_k\widehat{g}_k^*{\mathcal R}_1\times\cdots\times b_k\widehat{g}_k^*{\mathcal R}_l$ for all $k=1,2,\ldots,s$. Hence ${\mathcal R}=b_1\widehat{g}_1^*{\mathcal R}+\cdots+b_s\widehat{g}_s^*{\mathcal R}$. Let $c_1$, $c_2$, $\ldots$, $c_s\in {\mathcal R}$ satisfying $b_1\widehat{g}_1^*c_1+\cdots+b_s\widehat{g}_s^*c_s=0$. Since $(x^{m_i}-1)\mid g^*$ for all $i=1,2,\ldots,l$, it follows that $g^*{\mathcal R}=\{0\}$. Then for each $k=1,2,\ldots,s$, from $b_k\widehat{g}_k^*+s_kg_k^*=1$, $g^*=g_k^*\widehat{g}_k^*$ and $g^*\mid \widehat{g}_\tau^* \widehat{g}_\sigma^*$ for all $1\leq \tau\neq \sigma \leq s$, we deduce $b_k\widehat{g}_k^*c_k=0$. Hence ${\mathcal R}=\bigoplus_{j=1}^sb_j\widehat{g}_j^*{\mathcal R}$.
Define $\phi:~\beta_1+\beta_2+\cdots+\beta_s\mapsto (\mu_1(\beta_1), \mu_2(\beta_2), \ldots, \mu_s(\beta_s))~\mbox{where}\; \beta_k\in b_k\widehat{g}_k^*{\
| 2,836
| 1,430
| 1,946
| 2,591
| 4,108
| 0.768053
|
github_plus_top10pct_by_avg
|
ay". If you split a small vim window but still with a long message bar, the hit-enter prompt won't occur.
A:
Is there some way to totally avoid hit-enter prompts that are caused only by "not enough message space"?
No. Only "more prompts" can be fully disabled. See :help hit-enter.
I believe the hit-enter prompt here was caused by the too long message when executing the search command.
I guess, not the length but the number of messages: one message for echoing the mapping, and another for / output.
Try to use silent like this:
onoremap <silent>i@ :<c-u>silent! execute ...
The first <silent> suppresses "echo", the second one suppresses the command's output.
Q:
How to restrict access to the site during development?
I am creating a new Drupal 7 based site.
The development will be on a server that is publicly accessible. I am working in a multi-site environment.
I would like to totally block access to the site to anyone & anything but authorized users. Including access to the site name, theme, etc...
I used Secure Site for similar tasks before. It did http-level authentication and returned 403 when authentication failed. It doesn't have a D7 version.
What would be the easiest way to do this?
A:
Make a module and paste following code in your module file:
<?php
/**
* Implementation of hook_boot().
*
* Ask for user credentials and try to authenticate.
*/
function foo_boot() {
require_once DRUPAL_ROOT . '/includes/password.inc';
if (isset($_SERVER['PHP_AUTH_USER']) && isset($_SERVER['PHP_AUTH_PW'])) {
$query = "SELECT pass FROM {users} WHERE name = :name";
$result = db_query($query, array(':name' => $_SERVER['PHP_AUTH_USER']));
$account = new stdClass();
foreach ($result as $row) {
$account->pass = $row->pass;
}
if (isset($account->pass)) {
if (user_check_password($_SERVER['PHP_AUTH_PW'], $account)) {
return;
}
}
}
header('WWW-Authenticate: Basic realm="Development"');
header('HTTP/1.0 401 Unauthorized');
exit;
}
This uses HTTP Aut
| 2,837
| 670
| 1,944
| 2,269
| 51
| 0.831458
|
github_plus_top10pct_by_avg
|
sible.
[*(iv)*]{} Suppose that $m(P)=2$ and $m(Q)=1$. Then $P = P_r + P_p$ and $Q=Q_q$. By Lemma \[a20Sep16\] the equality $[P,Q]=1$ implies that $[P_p, Q_q]=0$ and $[P_r, Q_q]=1$. Then, $q\geq 0$, by Lemma \[a20Sep16\]. The case $q=0$ is not possible since then both $P_r, Q_q\in K[H]$ and this would contradict the equality $[P_r, Q_q]=1$. Therefore, $q>0$. Then $P_r = \beta Y^q$ and $Q_q = \alpha X^q$ for some nonzero elements $\beta, \alpha \in K[H]$. Then $$-1 = [Q_q, P_r] = (1-\s^{-q}) (\alpha \s^p(\beta) (q,-q) )$$ implies that $0 = \deg (-1) = \deg\, (1-\s^{-q}) (\alpha \s^p(\beta) (q,-q) ) = \deg \alpha + \deg \beta +q -1$, by Equation (\[degsf\]). Hence, $q=1$, $\alpha, \beta \in K^*$ and $\beta = -\alpha^{-1}$. Then $P, Q \in A_{1,\leq 1}$, and, by the statement (i), the pair $(P,Q)$ is obtained from the pair $(Y,X)$ by applying an automorphism of $A_1$. [*(v)*]{} $(m(P), m(Q)) \neq (2,2)$: Since $m(P)=m(Q)=2$, we can write $P = P_r + P_p$ and $Q = Q_s + Q_q$ as sums of homogeneous elements where $r < p$, $P_r \in A_{1,r}$, $P_p \in A_{1,p}$ and $s<q$, $Q_s \in A_{1,s}$, $Q_q \in A_{1,q}$. The equality $[P,Q]=1$ implies that $[P_r, Q_s]=0$ and $[P_p, Q_q]=0$ (see Lemma \[a20Sep16\]). By Lemma \[a20Sep16\], the elements $r$ and $s$ have the same sign (i.e., either $r<0, s<0$ or $r=s=0$ or $r>0, s>0$) and also the elements $p$ and $q$ have the same sign. Since $p\geq 2$, we must have $q>0$.
Suppose that $r\geq 0$, we seek a contradiction. Then $s\geq 0$ and so the elements $P$ and $Q$ are elements of the subring $A_{1,+}= \oplus_{i \geq 0} K[H] X^i$. Now, $$K[H] \ni 1 = [P, Q] \in [A_{1,+}, A_{1,+}] \subseteq \oplus_{i \geq 1} K[H] X^i,$$ a contradiction. Therefore, $r<0$ and $s<0$.
The equality $1=[P,Q]=[P_r, Q_q] + [P_p, Q_s]$ and Lemma \[a20Sep16\] imply that $r+q=0$ and $p+s=0$, that is $r=-q$ and $s=-p$. So, $$P = P_{-q} + P_p \text{ and } Q = Q_{-p} + Q_q.$$ The elements $P_p$ and $P_{-q}$ are homogeneous elements of the Weyl algebra $A_1$. The Weyl algebra $A_1$ is a homogeneous subalgebra of the
| 2,838
| 2,147
| 2,667
| 2,551
| 4,021
| 0.768608
|
github_plus_top10pct_by_avg
|
dependent set of data). As in the mentioned papers, expressions such as (\[abr1\]) will be referred to here as ‘ideal’ estimators.
It was once believed that $f_t$ in (\[abr1\]) could be replaced by $f$, but Terrell and Scott (1992) showed that in this case the bias reduction at a single $t$ depends heavily on the tail of $f$ and becomes negligible in the normal case (see also Hall, Hu and Marron (1995) and McKay (1993)). Taking $f_t$ instead of $f$ as Abramson did constitutes a way to deal with the tail effects on the localities $t$. Hall, Hu and Marron (1995), McKay (1993) and Novak (1999) also devised other ways of dealing with the problem. In particular, Hall, Hu and Marron proposed the ideal estimator $$\label{ideal0}
\bar f_n(t)=\frac{1}{n h_n}\sum_{i=1}^{n}K\left(\frac{t-X_i}{h_n} f^{1/2}(X_i)\right) f^{1/2}(X_i)I(|t-X_{i}|<h_nB),$$ for some $B>0$. Novak replaces $h_n$ in the indicator by $h_n/f^{1/2}(t)$ and considers powers other than 1/2 as well, and McKay replaces $ f^{1/2}_t(x)$ in (\[abr1\]) by a smooth function $\alpha (x) =cv^{1/2}(f(x)/c^2)$ with $v(t)=t$ for all $t\ge t_0\ge 1$ with the first four derivatives of $v$ vanishing at zero. We will focus our attention only on the simplest of these ideal estimators, which is (\[ideal0\]), although our results should hold for the other versions as well. The ideal estimator will only be a means to study the ‘true’ estimator, obtained from the ideal by replacement of $f$ by a preliminary estimator.
Specifically, in this article we study the uniform approximation of a density $f$ by estimators of the form $$\label{realest0}
\hat f(t;h_{1,n}, h_{2,n})=\frac{1}{n h_{2,n}}\sum_{i=1}^{n}K\left(\frac{t-X_i}{h_{2,n}}\hat f^{1/2}(X_i;h_{1,n})\right)\hat f^{1/2}(X_i;h_{1,n})I(|t-X_{i}|<h_{2,n}B),$$ where $\hat f(x;h_{1,n})$ is the classical kernel density estimator $$\hat f(x;h_{1,n})=\frac{1}{n h_{1,n}}\sum_{i=1}^{n}K\left(\frac{x-X_i}{h_{1,n}}\right)$$ and $h_{i,n}$ are two sequences of bandwidths that tend to zero as $n\to\infty$. Ideally, we would lik
| 2,839
| 2,851
| 2,687
| 2,660
| 1,931
| 0.784305
|
github_plus_top10pct_by_avg
|
ith the norm $(4)$ (so that it is isometric to $H(1)$ by Theorem \[210\]). Then by observing a formal matrix description of an element of $\underline{M}(R)$ for a $\kappa$-algebra $R$, explained in Section \[m\], the above formal matrix turns to be $$\begin{pmatrix}1-2z_j&0&0&0\\\pi z_j&1&0&\pi z_j\\\pi z_j&0&1&0\\\pi x_j+2 z_j&0&0&1 \end{pmatrix}.$$
We now follow the argument of the first case dealing with $j-1$ even. Namely, for a Jordan splitting $Y(C(L^{j-1}))=\oplus_{i\geq 0}M_i''$, we describe the image of a fixed element of $F_j$ in the orthogonal group associated to $M_0''$. There are three cases depending on the types of $M_0$.\
1. Assume that $M_0$ is *of type II*. In this case, $$M_0''=Be_1'\oplus (\pi)e_2'\oplus M_2\oplus \pi M_0.$$ Since $M_0''$ is *free of type II*, the image of a fixed element of $F_j$ in the orthogonal group associated to $M_0''$ is $$\begin{pmatrix}1&0&0\\ (z_j)_1&1&0\\0&0&id\\ \end{pmatrix}.$$ Here, $z_j=(z_j)_1+\pi \cdot (z_j)_2$ and $id$ is associated to $M_2\oplus \pi M_0$. Then the Dickson invariant of the above matrix is $(z_j)_1$.
In conclusion, $(z_j)_1$ is the image of a fixed element of $F_j$ under the map $\psi_j$. Since $(z_j)_1$ can be either $0$ or $1$ by Equation (\[e42\]), $\psi_j|_{F_j}$ is surjective onto $\mathbb{Z}/2\mathbb{Z}$ and thus $\psi_j$ is surjective.\
2. Assume that $M_0$ is *of type $I^e$*. Let $M_0=\left(\oplus H(0)\right)\oplus A(1, 2a, 1)$ and let $(e_5, e_6)$ be a basis for $A(1, 2a, 1)$. We consider the lattice spanned by $(e_5, e_6, e_1', e_2')$ with the Gram matrix $A(1, 2a, 1)\oplus A(4(b+b'), -2\delta(1+4b'), \pi(1+4b'))$. Then by Theorem \[210\], there is a suitable basis for this lattice such that the norm of the $\pi^1$-modular Jordan component is the ideal $(4)$. Namely, we choose $$(e_5-e_1', e_6-e_1', e_1'-\frac{2\pi(b+b') }{\delta(1+4b')}e_2', \pi e_5+\frac{1}{1+4b'}e_2').$$ Here, a method to find the above basis follows from the argument used in Case (iii) of Case (1) with $j$ even. Then the lattice spanned by the latter t
| 2,840
| 1,509
| 2,530
| 2,682
| null | null |
github_plus_top10pct_by_avg
|
an. J. Sci. Technol. **36**(A3) (2012) (Special Issue-Mathematics), 371-376.
M. Mursaleen, A.K. Noman, Applications of Hausdorff measure of noncompactness in the spaces of generalized means, Math. Inequal. Appl. **16** (2013) 207-220.
M. Stieglitz, H. Tietz, Matrix transformationen von folgenräumen eine ergebnisübersicht, Math. Z. **154** (1977) 1–16.
---
abstract: |
We consider evolution equations of the form $$\label{Abstract equation}
\dot u(t)+{\mathcal{A}}(t)u(t)=0,\ \ t\in[0,T],\ \
u(0)=u_0,$$ where ${\mathcal{A}}(t),\ t\in [0,T],$ are associated with a non-autonomous sesquilinear form ${\mathfrak{a}}(t,\cdot,\cdot)$ on a Hilbert space $H$ with constant domain $V\subset H.$ In this note we continue the study of fundamental operator theoretical properties of the solutions. We give a sufficient condition for norm-continuity of evolution families on each spaces $V, H$ and on the dual space $V'$ of $V.$ The abstract results are applied to a class of equations governed by time dependent Robin boundary conditions on exterior domains and by Schrödinger operator with time dependent potentials.
address:
- 'University of Wuppertal School of Mathematics and Natural Sciences Arbeitsgruppe Funktionalanalysis, Gaußstraße 20 D-42119 Wuppertal, Germany'
- ' Ibn Zohr University, Faculty of Sciences Departement of mathematics, Agadir, Morocco'
author:
- 'Omar EL-Mennaoui and Hafida Laasri\'
title: 'On the norm-continuity for evolution family arising from non-autonomous forms$^*$[^1]'
---
Introduction\[s1\] {#introductions1 .unnumbered}
==================
Throughout this paper $H,V$ are two separable Hilbert spaces over $\mathbb C$ such that $V$ is densely and continuously embedded into $H$ (we write $V \underset d \hookrightarrow H$). We denote by $(\cdot {\, \vert \,}\cdot)_V$ the scalar product and $\|\cdot\|_V$ the norm on $V$ and by $(\cdot {\, \vert \,}\cdot)_H, \|\cdot\|_H$ the corresponding quantities in $H.$ Let $V'$ be the antidual of $V$ and denote by $\langle ., . \rangle$ the duality betw
| 2,841
| 1,751
| 566
| 2,701
| 3,041
| 0.775282
|
github_plus_top10pct_by_avg
|
plane. Overall, the numerical results are exceedingly close to the analytic potential, with the mean relative error less than 0.1%. The errors are largest near the sphere boundary whose exact shape is not well resolved by any of the adopted grids.
[cccccc]{}\[!t\] Cartesian & $[-0.5,0.5]\times[-0.5,0.5]\times[-0.5,0.5]$ & $64\times 64\times 64$ & $0.25$ & $0.1$ & $-0.04$\
uniform cylindrical & $[0.5,1]\times[0,2\pi]\times[-0.25,0.25]$ & $64\times 256\times 64$ & $0.72$ & $0.63$ & $-0.04$\
logarithmic cylindrical & $[10^{-2},1]\times[0,2\pi]\times[-0.25,0.25]$ & $128\times 64\times 64$ & $0.27$ & $0.38$ & $-0.04$
Convergence Test {#s:convergence}
----------------
The discrete Poisson equation used for the interior solver in Section \[s:interior\_solver\] and for the boundary condition in Section \[s:bc\] are second-order accurate by construction. If our implementation of the Poisson solver is correct, therefore, the relative errors should be inversely proportional to the square of the grid spacing. To check if this is indeed the case, we repeat the uniform sphere tests by varying the number of cells from $16^3$ to $512^3$. Figure \[fig:convergence\] plots as circles the mean relative errors $\left\langle {\epsilon} \right\rangle$ from the sphere tests as functions of $N_z$. Overall, the errors decrease roughly at a second-order rate with increasing $N_z$, but exhibit some fluctuations. @katz16 noted that these fluctuations of the errors are caused not by the truncation errors of the finite-difference scheme but by inability of an adopted grid to perfectly resolve a spherical mass distribution. This is true even for a spherical grid when the sphere center offsets from the origin.
To delineate the truncation errors alone, it is thus necessary to design a solid figure whose shape is identical to the cell shape of an adopted grid. In addition, the size and mass of the solid figure should be unchanged with varying resolution. For this purpose, we consider a uniform cube with density $\rho=1$, located at $x_1 \le x
| 2,842
| 584
| 3,043
| 2,801
| null | null |
github_plus_top10pct_by_avg
|
tations by computing adversarial samples for a CNN. In particular, influence functions [@CNNInfluence] were proposed to compute adversarial samples, provide plausible ways to create training samples to attack the learning of CNNs, fix the training set, and further debug representations of a CNN. [@banditUnknown] discovered knowledge blind spots (unknown patterns) of a pre-trained CNN in a weakly-supervised manner.
Zhang *et al.* [@CNNBias] developed a method to examine representations of conv-layers and automatically discover potential, biased representations of a CNN due to the dataset bias. Furthermore, [@wu2007compositional; @yang2009evaluating; @wu2011numerical] mined the local, bottom-up, and top-down information components in a model for prediction.
**CNN semanticization:**[` `]{} Compared to the diagnosis of CNN representations, semanticization of CNN representations is closer to the spirit of building interpretable representations.
Hu *et al.* [@LogicRuleNetwork] designed logic rules for network outputs, and used these rules to regularize neural networks and learn meaningful representations. However, this study has not obtained semantic representations in intermediate layers. Some studies extracted neural units with certain semantics from CNNs for different applications. Given feature maps of conv-layers, Zhou *et al.* [@CNNSemanticDeep; @CNNSemanticDeep2] extracted scene semantics. Simon *et al.* mined objects from feature maps of conv-layers [@ObjectDiscoveryCNN_2], and learned explicit object parts [@CNNSemanticPart].
Unlike above research, we aim to explore the entire semantic hierarchy hidden inside conv-layers of a CNN. Because the AOG structure [@MumfordAOG; @MiningAOG] is suitable for representing the semantic hierarchy of objects, our method uses an AOG to represent the CNN. In our study, we use semantic-level QA to incrementally mine object parts from the CNN and grow the AOG. Such a “white-box” representation of the CNN also guided further active QA. With clear semantic struc
| 2,843
| 909
| 3,238
| 2,576
| null | null |
github_plus_top10pct_by_avg
|
anisotropic power spectra.
Finally, putting eq. (\[distortion\]) into eq. (\[projection\]), it can be seen that the one-dimensional redshift-space power spectrum is related to the isotropic three-dimensional real-space power spectrum by a linear integral equation: $$P (k_{\parallel}) = \int_{k_{\parallel}}^\infty
W(k_{\parallel}/k,k) {\tilde P} (k) {k dk
\over {2 \pi}}
\label{projection2}$$
Thus far, we have not specified the actual random field whose power spectrum we are interested in. The random field could be the mass overdensity $\delta =
\delta\rho/\bar\rho$ or the transmission/flux overdensity $\delta_f =
\delta f/\bar
f$, where $f = e^{-\tau}$, $\bar f = \langle f \rangle$, $\delta f = f - \bar f$, and $\tau$ is the optical depth. We will use $P^\rho$ or ${\tilde P}^\rho$ to denote the mass power spectrum and $P^{f}$ or ${\tilde P}^f$ to denote the transmission power spectrum.
The one-dimensional redshift-space transmission power spectrum can also be related to the three-dimensional real-space mass power spectrum by an effective kernel, which we will call $W^{f\rho}$: $$P^f (k_{\parallel}) = \int_{k_{\parallel}}^\infty
W^{f\rho}(k_{\parallel}/k,k) {\tilde P}^\rho (k) {k dk
\over {2 \pi}}
\label{projection2b}$$
In discretized form, this is equivalent to: $${\bf P^f } = {\bf A} \cdot {\bf {\tilde P^\rho}}
\label{projection3}$$ where the power spectra are represented as vectors and ${\bf A}$ is an upper (or lower) triangular matrix, which is invertible if none of the diagonal entries of ${\bf A}$ vanishes. The special case considered by Croft et al. [-@croft98] corresponds to $W^{f\rho} = \, {\rm const.}$, where inverting the above matrix equation is equivalent to the differentiation of $P^f (k_{\parallel})$.
The problem of eq. (\[projection3\]) is of course that $P^f
(k_{\parallel})$, for any given $k_{\parallel}$, depends on an infinite vector: ${\tilde P^\rho}
(k)$ for all $k$’s, from $k_{\parallel}$ to, in principle, $\infty$. To make it useful for computation, we have to truncate the infini
| 2,844
| 4,284
| 2,910
| 2,665
| 3,498
| 0.771923
|
github_plus_top10pct_by_avg
|
---------------
The observable matter states in heterotic–string vacuum with $(2,2)$ world–sheet supersymmetry is embedded in the $\bf{27}$ representation of $E_6$. In the free fermionic construction that we adopt here, and using the basis vectors in (\[421\]), the $E_6$ is first broken to the $SO(10)\times U(1)$ symmetry. Therefore, the $\bf{27}$ of $E_6$ decomposes in the following way $$\begin{aligned}
\textbf{27} &= & \textbf{16} + \textbf{10} + \textbf{1}.\end{aligned}$$ Where the $\textbf{16}$ transforms under the spinorial representation of $SO(10)$ and **10** transforms in the vectorial representation of the $SO(10)$, and similarly for $\bf{\overline{27}}$. The following 48 sectors produce states that give the spinorial $\bf{16}$ or $\bf{\overline{16}}$ of $SO(10)$ $$\begin{aligned}
\label{obspin}
B_{pqrs}^{(1)}&=& S + {b_1 + p e_3+ q e_4 + r e_5 + s e_6} \nonumber\\
&=&\{\psi^\mu,\chi^{12},(1-p)y^{3}\overline{y}^3,p\omega^{3}\overline{\omega}^3,
(1-q)y^{4}\overline{y}^4,q\omega^{4}\overline{\omega}^4, \nonumber\\
& & ~~~(1-r)y^{5}\overline{y}^5,r\omega^{5}\overline{\omega}^5,
(1-s)y^{6}\overline{y}^6,s\omega^{6}\overline{\omega}^6,
\overline{\eta}^1,\overline{\psi}^{1,...,5}\},
\\
B_{pqrs}^{(2)}&=& S + {b_2 + p e_1+ q e_2 + r e_5 + s e_6},
\label{twochiralspinorials}
\nonumber\\
B_{pqrs}^{(3)}&=& S + {b_3 + p e_1+ q e_2 + r e_3 + s e_4}, \nonumber\end{aligned}$$ where $p,q,r,s=0,1$ and $b_3=b_1+b_2+x$. In order to distinguish between the spinorial $\bf{16}$ and $\bf{\overline{16}}$ in the states given above, the following chirality operators are used
$$\begin{aligned}
\label{so10operators}
X_{pqrs}^{(1)_{SO(10)}} & = &
C\binom{B^{(1)}_{pqrs}}{b_{2} + (1-r)e_{5} + (1-s)e_{6}},\nonumber\\
X_{pqrs}^{(2)_{SO(10)}} & = &
C\binom{B^{(2)}_{pqrs}}{b_{1} + (1-r)e_{5} + (1-s)e_{6}},\\
X_{pqrs}^{(3)_{SO(10)}} & = &
C\binom{B^{(3)}_{pqrs}}{b_{1} + (1-r)e_{3} + (1-s)e_{4}}.\nonumber\end{aligned}$$
Where $X_{pqrs}^{(1,2,3)_{SO(10)}} = 1$ implies the states corresponds to the $\bf{16}$ of $SO(10)$ and $X_{pqrs}
| 2,845
| 1,105
| 1,777
| 2,866
| 3,134
| 0.774625
|
github_plus_top10pct_by_avg
|
15 Continuous HeartMate II RIFLE criteria AKI\
1/15 (6.7%)
Slaughter et al. \[[@CIT0058]\] 2013 USA Patients underwent LVAD implantation as a bridge to transplant in 2008 332 Continuous\ INTERMACS criteria:\ AKI:\
HeartWare RRT or an increase in SCr ≥3 times baseline or SCr ≥ 5 mg/dL sustained for over 48 hours 32/332 (9.6%)
Tsiouris et al. \[[@CIT0066]\] 2013 USA Patients underwent LVAD implantation during 200
| 2,846
| 4,913
| 2,104
| 2,044
| null | null |
github_plus_top10pct_by_avg
|
s exponentially with the chain order and the available data is too sparse for proper parameter inferences. Thus, we show further evidence that the memoryless model seems to be a quite practical and legitimate model for human navigation on a page level.
- By abstracting away from the page level to a topical level, the results are different. By representing all datasets as navigational sequences of topics that describe underlying Web pages (cf. Figure \[fig:pathexample\]), we find evidence that topical navigation of humans is not memoryless at all. On three rather different datasets of navigation – free navigation (MSNBC) and goal-oriented navigation (WikiGame and Wikispeedia) – we find mostly consistent memory regularities on a topical level: In all cases, Markov chain models of order two (respectively three) best explain the observed navigational sequences. We analyze the structure of such navigation, identify strategies and the most salient common sequences of human navigational patterns and provide visual depictions. Amongst other structural differences between goal-oriented and free form navigational patterns, users seem to stay in the same topic more frequently for our free form navigational dataset (MSNBC) compared to both of the goal oriented datasets (Wikigame and Wikispeedia). Our analysis thereby provides new insights into the memory and structure that users employ when navigating the Web that can e.g., be useful to improve recommendation algorithms, web site design or faceted browsing.
The paper is structured as follows: In the section entitled “” we review the state-of-the-art in this domain. Next, we present our methodology and experimental setup in the sections called “” and “”. We present and discuss our results in the section named “”. In the section called “ we provide a final discussion and the section called ”" concludes our paper.
Related Work {#sec:related .unnumbered}
============
In the late 1990s, the analysis of user navigational behavior on the Web became an
| 2,847
| 2,061
| 3,073
| 2,031
| null | null |
github_plus_top10pct_by_avg
|
sition \[magneticeigenbladeterminant\], below, yields $f_1\equiv f_2\equiv0$, which gives $\ker\left({\mathbf{N}}_2\right)=\{0\}$. $\blacksquare$
Now we want to determine the prefactor in Equation . Recall that the determinant of a diagonalizable operator is defined as the product of its eigenvalues, if it exists. We have the following proposition.
\[magneticeigenbladeterminant\] Let ${\mathbf{K}}$ be as in , ${\mathbf{L}}$ as in Proposition \[magneticL\]. Then
- For ${\mathbf{L}}({\mathbf{Id}}+{\mathbf{K}})^{-1}:L^2_{2}({\mathbb{R}},dx)_{{\mathbb{C}}}{\rightarrow}L^2_{2}({\mathbb{R}},dx)_{{\mathbb{C}}}$, the non-vanishing eigenvalues and their corresponding eigenvectors are $$\begin{aligned}
\lambda_n&=\frac{2k}{(2n-1)\pi}t,\\
e_n(\cdot)&=c_1\left(\begin{matrix}
{\mathbf{1}}_{[0,t)}(\cdot)\cos\left(\frac{2k}{\lambda_n}\cdot\right)\\
{\mathbf{1}}_{[0,t)}(\cdot)\sin\left(\frac{2k}{\lambda_n}\cdot\right)
\end{matrix}\right)+c_2\left(\begin{matrix}
{\mathbf{1}}_{[0,t)}(\cdot)\sin\left(\frac{2k}{\lambda_n}\cdot\right)\\
-{\mathbf{1}}_{[0,t)}(\cdot)\cos\left(\frac{2k}{\lambda_n}\cdot\right)
\end{matrix}\right),\end{aligned}$$ $n\in{\mathbb{Z}},c_1,c_2\in{\mathbb{C}}$, where the multiplicity of the eigenvalues is 2.
- We have for the determinant $$\begin{aligned}
\det\left({\mathbf{Id}}+{\mathbf{L}}({\mathbf{Id}}+{\mathbf{K}})^{-1}\right)=\cos^2\!\left(kt\right).\end{aligned}$$
[**Proof:**]{} (i): We want to calculate the eigenvalues of $$\begin{aligned}
&{\mathbf{L}}({\mathbf{Id}}+{\mathbf{K}})^{-1}\\[3mm]
=&\left(\begin{matrix}
0 & ikP_{[0,t)}\left(A-A^*\right)P_{[0,t)}\\
ikP_{[0,t)}\left(A^*-A\right)P_{[0,t)} & 0\end{matrix}
\right)\\
&\hspace{55mm}\times
\left(\begin{matrix}
iP_{[0,t)}+P_{[0,t)^c}&0\vphantom{ikP_{[0,t)}\left(A-A^*\right)}\\
0\vphantom{ikP_{[0,t)}\left(A-A^*\right)}&iP_{[0,t)}+P_{[0,t)^c}
\end{matrix}\right)\\[5mm]
=&\left(\begin{matrix}
0&kP_{[0,t)}\left(A^*-A\right)P_{[0,t)}\\
kP_{[0,t)}\left(A-A^*\right)P_{[0,t)}&0
\end{matrix}\rig
| 2,848
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| 2,434
| 2,566
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|
bination of semistandard homomorphisms using three applications of Lemma \[lemma7\], from which it follows that ${\hat\Theta_{E}}\neq0$.
To prove that ${\hat\Theta_{T}}\circ{\hat\Theta_{C}}={\hat\Theta_{E}}$, use the notation of Proposition \[tabcomp\], with $S=C$. Suppose $X\in\calx$ is such that the coefficient of ${\hat\Theta_{U_X}}$ in Proposition \[tabcomp\] is non-zero. Since $X^{31}$ must be $\{2\}$, we cannot have $X^{21}=\{2\}$ (because this would give a factor $\binom21$), so $X^{21}=\{1\}$ and hence $X^{23}=\{1,2\}$. Now if $X^{11}$ contains any of the numbers $1,\dots,v$ then again we get a factor $\binom21$. So we have $X^{12}=\{1,\dots,v\}$, which determines $X$, and we find that ${\hat\Theta_{T}}\circ{\hat\Theta_{C}}={\hat\Theta_{E}}$ as required.
So we have $\sigma\circ{\hat\Theta_{C}}=|{\calu}|{\hat\Theta_{E}}$, which is non-zero if and only if $|{\calu}|$ is odd. But it is easy to see that $|{\calu}|=\mbinom{u-v}{a-v}$, and the proposition is proved.
Suppose $S^\mu=S^{(u,v,2)}$ is irreducible.
Suppose first that $\mbinom{u-v}{a-v}$ is odd. This implies in particular that $0\ls a-v\ls u-v$, so $v\ls\min\{a-1,b+1\}$. So the assumptions of this section are valid, and we have homomorphisms ${\hat\Theta_{C}}:S^\mu\to S^\la$ and $\sigma:S^\la\to S^{\mu'}$. By Proposition \[comp2\], $\sigma\circ{\hat\Theta_{C}}\neq0$, so $S^\mu$ is a summand of $S^\la$.
Conversely, suppose we have homomorphisms $S^\mu\stackrel\gamma\longrightarrow S^\la\stackrel\delta\longrightarrow S^{\mu'}$ with $\delta\circ\gamma\neq0$. By Propositions \[cd2homdim1\] and \[cd2homdim2\], $\delta$ must be a scalar multiple of $\sigma$, and $\gamma$ must be a scalar multiple of ${\hat\Theta_{C}}$. Hence by Proposition \[comp2\], $\mbinom{u-v}{a-v}$ is odd.
Decomposability of Specht modules {#whichdec}
=================================
In this section, we prove Corollary \[maincor\], which answers the question of which Specht modules are shown to be decomposable by Theorem \[main\]. First we consider the case where $a+b\equiv0\pp
| 2,849
| 1,327
| 1,166
| 2,669
| 3,940
| 0.769125
|
github_plus_top10pct_by_avg
|
named entities, geographical locations and temporal expressions. What would be the most descriptive mathematical models for each of these semantic annotations?
**Similarity Functions**. Given a pair of named entities, geographical locations or temporal expressions; how can we efficiently compute the similarity between the same type of annotations?
**Efficiency & Scalability**. Identifying data structures for indexing corpora along with their semantic annotations, such that their asymptotic run times scale linearly with the size of the corpora.
**Evaluation**. Since evaluation of the solutions outlined are very subjective in nature; what are other reliable sources of objective ground truth ? What other metrics can be employed to test the effectiveness of our methods ?
Conclusion
==========
\[sec:conclusion\]
In this article I laid out an outline of the research work that I envisage to carry out for my PhD dissertation. The research would in its culmination provide us methods to computationally extract world history as sequence of temporally ordered events and portray future events to take place from semantically annotated corpora. The research would also provide ways to perform semantic search and large scale event analytics on these annotated corpora. I further described already available resources that can be utilized for carrying out the research; test cases that can be built from encyclopedic resources on the Internet; and the metrics that can be utilized for evaluation.
[10]{} Clueweb’09. <http://www.lemurproject.org/clueweb09.php/>.
Clueweb’12. <http://www.lemurproject.org/clueweb12.php/>.
English gigaword. <https://catalog.ldc.upenn.edu/LDC2003T05>.
New york times anotated corpus. <https://catalog.ldc.upenn.edu/LDC2008T19>.
Abujabal A. and Berberich K. Important events in the past, present, and future. WWW’15 Companion Volume.
Agrawal R. et al. Diversifying search results. WSDM’09.
Allan J., editor. . Kluwer Academic Publishers, Norwell, MA, USA, 2002.
Baeza-Yates R. Searching the fu
| 2,850
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| 0.782061
|
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Hida distribution, see Theorem \[magnetictheorem\].
- The results in Theorem \[magnetictheorem\] provide us with the generating functional for a charged particle in a constant magnetic field.
- The generalized expectations (generating functional at zero) yields the Greens functions to the corresponding Schrödinger equation.
White Noise Analysis
====================
Gel’fand Triples
----------------
Starting point is the Gel’fand triple $S_d({\mathbb{R}}) \subset L^2_d({\mathbb{R}},dx) \subset S'_d({\mathbb{R}})$ of the ${\mathbb{R}}^d$-valued, $d \in {\mathbb{N}}$, Schwartz test functions and tempered distributions with the Hilbert space of (equivalence classes of) ${\mathbb{R}}^d$-valued square integrable functions w.r.t. the Lebesgue measure as central space (equipped with its canonical inner product $(\cdot, \cdot)$ and norm $\|\cdot\|$), see e.g. [@W95 Exam. 11]. Since $S_d({\mathbb{R}})$ is a nuclear space, represented as projective limit of a decreasing chain of Hilbert spaces $(H_p)_{p\in {\mathbb{N}}}$, see e.g. [@RS75a Chap. 2] and [@GV68], i.e. $$S_d({\mathbb{R}}) = \bigcap_{p \in {\mathbb{N}}} H_p,$$ we have that $S_d({\mathbb{R}})$ is a countably Hilbert space in the sense of Gel’fand and Vilenkin [@GV68]. We denote the inner product and the corresponding norm on $H_p$ by $(\cdot,\cdot)_p$ and $\|\cdot\|_p$, respectively, with the convention $H_0 = L^2_d({\mathbb{R}}, dx)$. Let $H_{-p}$ be the dual space of $H_p$ and let $\langle \cdot , \cdot \rangle$ denote the dual pairing on $H_{p} \times H_{-p}$. $H_{p}$ is continuously embedded into $L^2_d({\mathbb{R}},dx)$. By identifying $L_d^2({\mathbb{R}},dx)$ with its dual $L_d^2({\mathbb{R}},dx)'$, via the Riesz isomorphism, we obtain the chain $H_p \subset L_d^2({\mathbb{R}}, dx) \subset H_{-p}$. Note that $\displaystyle S'_d({\mathbb{R}})= \bigcup_{p\in {\mathbb{N}}} H_{-p}$, i.e. $S'_d({\mathbb{R}})$ is the inductive limit of the increasing chain of Hilbert spaces $(H_{-p})_{p\in {\mathbb{N}}}$, see e.g. [@GV68]. We denote the
| 2,851
| 1,023
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| 2,704
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he state of the system and the eigenmode amplitudes of Ref. [@Terry2006] can be recovered. Consequently, the governing Eq. can be manipulated to derive nonlinear equations that describe the evolution of the eigenmode amplitudes and their interactions. The method relies on the jump conditions given in Eq. . Since the jump conditions for one eigenmode differ from those for the other eigenmode, one can form an invertible map between the discontinuity of $d\phi/dz$ at each interface and the amplitude of each eigenmode. Additionally, because there are two jump conditions that will serve as our dynamical quantities, only the two eigenmodes of the previous section are needed to construct an invertible map between eigenmodes and dynamical quantities. To derive equations describing the nonlinear interaction between the eigenmodes, we start by deriving nonlinear jump conditions.
First, let $\hat{\phi}(k,z,t) = \mathcal{F}[\Phi(x,z,t)]$ be the Fourier transformed stream function, and assume $$\label{phihat combo}
\hat{\phi}(k,z,t) = \beta_1(k,t)\phi_1(k,z) + \beta_2(k,t)\phi_2(k,z).$$ The nonlinear jump conditions are obtained by performing the same steps that led to Eq. without dropping nonlinear terms (and explicitly taking the Fourier transform rather than assuming normal modes). Taking the Fourier transform and integrating from $\pm 1 - \epsilon$ to $\pm 1 + \epsilon$ with $\epsilon \to 0$ yields $$\label{deltahat}
\frac{\partial}{\partial t}\hat{\Delta}_{\pm} \pm ik\hat{\Delta}_{\pm} \pm ik\hat{\phi}(k,\pm 1) + \lim\limits_{\epsilon\to 0}ik\int \limits_{-\infty}^{\infty}\frac{dk'}{2\pi}\left[\frac{d}{dz}\hat{\phi}(k',z)\frac{d}{dz}\hat{\phi}(k'',z)\right]_{\pm 1-\epsilon}^{\pm 1+\epsilon} = 0,$$ where $k'' \equiv k-k'$, while $$\begin{aligned}
\hat{\Delta}_{\pm}(k,t) &\equiv \lim\limits_{\epsilon\to 0}\left[ \frac{d}{dz}\hat{\phi}(k,\pm 1 + \epsilon,t) - \frac{d}{dz}\hat{\phi}(k,\pm 1 - \epsilon,t)\right]\\
&= \beta_1(k,t)\Delta_{\pm 1}(k) + \beta_2(k,t)\Delta_{\pm 2}(k)\end{aligned}$$ and $$\Delta_{\pm j}(k) \e
| 2,852
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| 3,105
| 2,715
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|
1^{+0.049}_{-0.048}$ from [@WMAP]. We also calculate $\Omega_{\hbox{\scriptsize asymp}}$, the fractional density of matter that *cannot* be determined through gravity, to be $0.197_{\pm 0.017}$, which is nearly equal to the fractional density of nonbaryonic matter $\Omega_m-\Omega_{B} =
0.196^{+0.025}_{-0.026}$ [@WMAP]. We then find the fractional density of matter in the universe that can be determined through gravity, $\Omega_{\hbox{\scriptsize Dyn}}$, to be $0.041^{+0.030}_{- 0.031}$, which is nearly equal to $\Omega_B=0.0416^{+0.0038}_{-0.0039}$. Details of our calculations and theory is in [@ADS].
Extending the GEOM and Galactic Structure
=========================================
Any extension of the geodesic action requires a dimensionless, scalar function of some property of the spacetime folded in with some physical property of matter. While before no such properties existed, with the discovery of Dark Energy there is now $\lambda_{DE}$ and these extensions can be made. As we work in the nonrelativistic, linearized gravity limit, we consider the simplest extension: $$\mathcal{L}_{\hbox{\scriptsize{Ext}}} =
mc\Big(1+\mathfrak{D}\left[Rc^2/ \Lambda_{DE}G\right]\Big)^{\frac{1}{2}}
\left(g_{\mu\nu}\frac{d x^\mu}{dt}\frac{d x^\nu}{dt}\right)^{\frac{1}{2}}
\equiv mc\mathfrak{R}[Rc^2/\Lambda_{DE}G] \left(g_{\mu\nu}\frac{d x^\mu}{dt}\frac{d x^\nu}{dt}\right)^{\frac{1}{2}}
\label{extendL}$$ with the constraint $v^2=c^2$ for massive test particles. Here, $\mathfrak{D}(x)$ is a function function given below, and $R$ is the Ricci scalar. For massive test particles, the extended GEOM is $v^\nu\nabla_\nu v^\mu = c^2\left(g^{\mu\nu} - v^\mu
v^\nu/c^2\right)\nabla_\nu \log\mathfrak{R}[4+8\pi
T/\Lambda_{DE}c^2]$, where $v^\mu$ is the four-velocity of a test particle, $T_{\mu\nu}$ is the energy-momentum tensor, $T=T_\mu^\mu$, and we take $\Lambda_{DE}$ to be the cosmological constant. As the action for gravity+matter is a linear combination of the Hilbert action and the action for matter, any changes to the equa
| 2,853
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t ${\mathrm{Hom}}(a,b)$ consists of the triples $(b,f,a)$, where $$f={\sigma }_{i_n}^{{r}_{i_{n-1}}\cdots {r}_{i_1}(a)}\cdots
{\sigma }_{i_2}^{{r}_{i_1}(a)}{\sigma }_{i_1}^a$$ and $b={r}_{i_n}\cdots {r}_{i_2}{r}_{i_1}(a)$ for some $n\in {\mathbb{N}}_0$ and $i_1,\ldots ,i_n\in I$. The composition is induced by the group structure of ${\mathrm{Aut}}({\mathbb{Z}}^I)$: $$(a_3,f_2,a_2)\circ (a_2,f_1,a_1) = (a_3,f_2f_1, a_1)$$ for all $(a_3,f_2,a_2),(a_2,f_1,a_1)\in {\mathrm{Hom}}({\mathcal{W}}({\mathcal{C}}))$. By abuse of notation we will write $f\in {\mathrm{Hom}}(a,b)$ instead of $(b,f,a)\in {\mathrm{Hom}}(a,b)$.
The cardinality of $I$ is termed the *rank of* ${\mathcal{W}}({\mathcal{C}})$. A Cartan scheme is called *connected* if its Weyl groupoid is connected.
Recall that a groupoid is a category such that all morphisms are isomorphisms. The Weyl groupoid ${\mathcal{W}}({\mathcal{C}})$ of a Cartan scheme ${\mathcal{C}}$ is a groupoid, see [@p-CH08]. For all $i\in I$ and $a\in A$ the inverse of ${\sigma }_i^a$ is ${\sigma }_i^{r_i(a)}$. If ${\mathcal{C}}$ and ${\mathcal{C}}'$ are equivalent Cartan schemes, then ${\mathcal{W}}({\mathcal{C}})$ and ${\mathcal{W}}({\mathcal{C}}')$ are isomorphic groupoids.
A groupoid $G$ is called *connected*, if for each $a,b\in {\mathrm{Ob}}(G)$ the class ${\mathrm{Hom}}(a,b)$ is non-empty. Hence ${\mathcal{W}}({\mathcal{C}})$ is a connected groupoid if and only if ${\mathcal{C}}$ is a connected Cartan scheme.
\[de:RSC\] Let ${\mathcal{C}}={\mathcal{C}}(I,A,({r}_i)_{i\in I},(C^a)_{a\in A})$ be a Cartan scheme. For all $a\in A$ let $R^a\subset {\mathbb{Z}}^I$, and define $m_{i,j}^a= |R^a \cap (\ndN_0{\alpha }_i + \ndN_0{\alpha }_j)|$ for all $i,j\in
I$ and $a\in A$. We say that $${\mathcal{R}}= {\mathcal{R}}({\mathcal{C}}, (R^a)_{a\in A})$$ is a *root system of type* ${\mathcal{C}}$, if it satisfies the following axioms.
1. $R^a=R^a_+\cup - R^a_+$, where $R^a_+=R^a\cap \ndN_0^I$, for all $a\in A$.
2. $R^a\cap {\mathbb{Z}}{\alpha }_i=\{{\alpha }_i,-{\alpha }_i\}$ for all
| 2,854
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|
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|
t $O$ randomly: $$\begin{gathered}
{{\mathbb P}}\big(\{\theta(i,j)\in A\}\cap \{M\geq 2\} \cap \Omega(i,j)\big)= \sum_{m=2}^5 {{\mathbb P}}\big(\{\theta(i,j)\in A\}\cap \{M=m\}\cap \Omega(i,j)\big)\\
\begin{aligned}
\leq & {{\mathbb P}}\big(\bigcup_{i\not= j\in \{1,\dots,m\}}\{\theta(\underline{i},\underline{j})\in A\}\cap \{M=m\}\cap \Omega(\underline{i},\underline{j})\big)\\
\leq & \sum_{m=2}^5 \sum_{i\not= j\in \{1,\dots,m\}}{{\mathbb P}}\big(\{\theta(\underline{i},\underline{j})\in A\}\cap \{M=m\}\big)\leq 0,
\end{aligned}\end{gathered}$$since $\lambda(A)=0$. Radon-Nikodym’s theorem concludes the proof. $\Box$
We conclude this section by a corollary that states that the asymptotic directions of competition interfaces and of semi-infinite paths are related.
The asymptotic direction of the competition interface $\varphi(i,j)$ belongs to the (random) set $D$ of directions with at least two semi-infinite paths. This set is a.s. dense in $[0,2\pi)$ and countable.
The fact that $D$ is dense in $[0,2\pi)$ follows from Part $(iii)$ of Theorem \[HN1\]. It is also a.s. countable. Indeed, let us consider the set $\Gamma$ of couples $(\gamma_{1},\gamma_{2})$ of different semi-infinite paths of the RST such that the region they delimit (in the trigonometric sense) contains only finite paths. Associating to each element $(\gamma_{1},\gamma_{2})$ of $\Gamma$ the child in $\gamma_{1}$ of their bifurcation point, we get an injective function from $\Gamma$ to the PPP $N$. Consequently, $\Gamma$ is a.s. countable. Moreover, Parts $(i)$ and $(ii)$ of Theorem \[HN1\] allow to associate to each element $(\gamma_{1},\gamma_{2})$ of $\Gamma$ their common asymptotic direction. This provides a surjective function from $\Gamma$ onto the set $D$. Hence $D$ is a.s. countable. $\Box$
Distribution of the $\theta(i,j)$’s and conjectures {#section:conjectures}
===================================================
In this section, we provide some clues and conjectures that may help understanding the distribution of the vector $(\the
| 2,855
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d], $$\begin{aligned}
\delta_{K\pi} &= \mathrm{arg}\left(- \frac{1-\frac{1}{2} \tilde{t}_1 }{1+\frac{1}{2} \tilde{t}_1 } \right)
= -\mathrm{Im} (\tilde{t}_1)\,, \label{eq:strongphase} \end{aligned}$$ where in the last step we neglect terms of relative order of $\varepsilon^2$.
After that we can determine $\tilde{s}_1$ and $\tilde{t}_2$ from Eqs. (\[eq:Res1tilde\]) and (\[eq:combi\]), respectively. The sum and difference of the integrated direct CP asymmetries can be used together with the phases $\delta_{KK}$ and $\delta_{\pi\pi}$ to determine $\tilde{p}_0$ and $\tilde{p}_1$. We have $$\begin{aligned}
\Delta a_{CP}^{\mathrm{dir}} &= \mathrm{Im}\left(\frac{\lambda_b}{\Sigma}\right) \times
4\,\mathrm{Im}\left(\tilde{p}_0 \right)
\,, \label{eq:DeltaACPdirParameter}\end{aligned}$$ and $$\begin{aligned}
\Sigma a_{CP}^{\mathrm{dir}} =
2\, \mathrm{Im}\left(\frac{\lambda_b}{\Sigma}\right)
\times \left[
2\, \mathrm{Im}(\tilde{p}_0 ) \tilde{s}_1 +
\mathrm{Im}(\tilde{p}_1) \right] \,.\end{aligned}$$ Note that also $\Delta a_{CP}^{\mathrm{dir}}$ and $\Sigma a_{CP}^{\mathrm{dir}}$ share the feature of corrections entering only at the relative order $\mathcal{O}(\varepsilon^2)$ compared to the leading result. The measurement of $\Delta a_{CP}^{\mathrm{dir}}$ is basically a direct measurement of $\mathrm{Im}\,\tilde{p}_0$, $$\begin{aligned}
\mathrm{Im}\,\tilde{p}_0 &= \frac{1}{4 \mathrm{Im}(\lambda_b/\Sigma)} \Delta a_{CP}^{\mathrm{dir}}\,. \label{eq:penguinovertree} \end{aligned}$$
The phases $\delta_{KK}$ and $\delta_{\pi\pi}$ give (see e.g. Ref. [@Nierste:2015zra]) $$\begin{aligned}
\mathrm{Re}\left(\frac{A_b(D^0\rightarrow K^+K^-)}{A_{\Sigma}(D^0\rightarrow K^+K^-)} \right) - \mathrm{Re}\left(\frac{A_b(D^0\rightarrow \pi^+\pi^-)}{A_{\Sigma}(D^0\rightarrow \pi^+\pi^-)} \right) &=
4 \mathrm{Re}(\tilde{p}_0) \,, \label{eq:retildep0}\end{aligned}$$ and $$\begin{aligned}
\mathrm{Re}\left(\frac{A_b(D^0\rightarrow K^+K^-)}{A_{\Sigma}(D^0\rightarrow K^+K^-)} \right) + \mathrm{Re}\left(\frac{A_b(D^0\rightarr
| 2,856
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of $S$ always has a positive lower bound in probability. Combing these three facts, it gives $$\|\lambda\|[ \theta^\top S_K\theta + O_{p}(K^{-1/2}) o_{p}(K^{1/2}) ] = O_{p}(K^{-1/2}).$$ So, we have $$\|\lambda\|= O_{p}(K^{-1/2}).$$ Furthermore, $$\label{eqA5}
\max_{1\leq k\leq K}|U_{km}|= O_{p}(K^{-1/2})o_{p}(K^{-1/2})=o_{p}(1).$$ Expanding (\[eqA3\]) gives $$\begin{aligned}
\label{eqA6}
0&=\frac{1}{K}\sum_{k=1}^{n} (Y_{km}- \mu) \Big{\{} 1 - U_{km} + \frac{U_{km}^{2}}{1 + U_{km}} \Big{\}} \notag \\
&=(\bar{Y}_{km}- \mu) - S_K \lambda + \frac{1}{K}\sum_{k=1}^{K} \frac{(Y_{km}- \mu)U_{km}^{2}}{1 + U_{km}}.\end{aligned}$$ The final term in (\[eqA6\]) above has a norm bounded by $$\frac{1}{K}\sum_{k=1}^{K}\|Y_{km}- \mu \| ^{3} \|\lambda\| ^2 |1+Y_{km}|^{-1} = o_{p}(K^{1/2})O_{p}(K^{-1}) O_{p}(1)=o_{p}(K^{-1/2}).$$ So, $$\lambda = S_K^{-1}(\bar{Y}_{km}-\mu) + \beta,$$ with $\beta =o_{p}(K^{-1/2})$. By (\[eqA5\]), we may expand $$\log\Big{(} 1+ U_{m,k} \Big{)}= U_{m, k} - \frac{1}{2}U_{m,k}^{2} + \eta_{k}$$ where for some finite $B >0, 1\leq k\leq K$, $${\mathbb{P}}(| \eta_{k}|\leq B|U_{km}|^3 )\to 1$$ as $K\to \infty$ and $m\to \infty$.
We can verify the follow the identities after some algebra $$\begin{aligned}
-2 \log \mathcal{R}(\mu) &=2\sum_{k=1}^{K}\log \Big{(} 1+ U_{km} \Big{)} \\
&= 2\sum_{k=1}^{K} \Big{(} U_{km} - \frac{1}{2}U_{km}^{2} + \eta_{k} \Big{)} \\
&=2K\lambda^\top(\bar{Y}_{Km} -\mu) -K\lambda^\top S_K\lambda + 2 \sum_{k=1}^{K} \eta_{i}\\
&= K(\bar{Y}_{Km} -\mu)^\top S_K^{-1} (\bar{Y}_{Km} -\mu) -K\beta^\top S_K^{-1}\beta + 2 \sum_{k=1}^{K} \eta_{k}.\end{aligned}$$ By Theorem \[theorem1\] and Lemma \[lem-2\] $$K(\bar{Y}_{km}-\mu)^\top S_K^{-1} (\bar{Y}_{km}-\mu) \stackrel{d}{\longrightarrow} \chi^{2}_{p}.$$ The second and third terms are $o_{p}(1)$ since $$K\beta^\top S_K^{-1}\beta=Ko_{p}(K^{-1/2})O_{p}(1)o_{p}(K^{-1/2})=o_{p}(1),$$ $$\Big{|} \sum_{k=1}^{K} \eta_{k} \Big{|}\leq B\|\lambda\|^3 \sum_{k=1}^{K} \| Y_{km}-\mu\|^{3} =O_{p}(K^{-3/2})o_{p}(K^{3/2})=o_{p}(1).$$
| 2,857
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M{l}$, their conjugate momenta $\MM{\pi}$ and the conjugate momentum $(\phi)$ to the advected quantities $(a)$ may be eliminated from equations (\[EL advected 1\]-\[EL advected 5\]) to obtain the weak form of the Euler-Poincaré equation with advected quantities: $${\frac{\partial }{\partial t}}{\frac{\delta \ell}{\delta \MM{u}}} + \operatorname{ad}^*_{\MM{u}}{\frac{\delta \ell}{\delta \MM{u}}}
= a\diamond{\frac{\delta \ell}{\delta a}}, \qquad a_t + \mathcal{L}_{\MM{u}}a = 0.$$
Take the time-derivative of the inner product of ${\delta\ell}/{\delta\MM{u}}$ with a function of $\MM{w}$: $$\begin{aligned}
{\frac{d }{d t}}\Bigg\langle {\frac{\delta \ell}{\delta \MM{u}}},\MM{w}\Bigg\rangle & = &
{\frac{d }{d t}}\Bigg\langle -(\nabla\MM{l})^T\cdot\MM{\pi} - \phi\diamond a
, \MM{w} \Bigg\rangle
= {\frac{d }{d t}}\Bigg\langle -\MM{\pi},(\MM{w}\cdot\nabla)\MM{l}\Bigg\rangle
+ {\frac{d }{d t}}\Bigg\langle \phi,\mathcal{L}_{\MM{w}}a\Bigg\rangle \\
& = & \Bigg\langle \nabla\cdot(\MM{u}\MM{\pi}),
(\MM{w}\cdot\nabla)\MM{l}\Bigg\rangle
+ \Bigg\langle \MM{\pi},
(\MM{w}\cdot\nabla)(\MM{u}\cdot\nabla)\MM{l}\Bigg\rangle
+ \Bigg\langle {\frac{\delta \ell}{\delta a}}\diamond a,\MM{w}\Bigg\rangle\\
& & \quad + \Bigg\langle -{\frac{\delta \ell}{\delta a}}-\mathcal{L}_{\MM{u}}\phi,
\mathcal{L}_{\MM{w}}a\Bigg\rangle
+ \Bigg\langle \phi,-\mathcal{L}_{\MM{w}}\mathcal{L}_{\MM{u}}a\Bigg\rangle
, \\
& = & \Bigg\langle \MM{\pi},
-(\MM{u}\cdot\nabla)(\MM{w}\cdot\nabla)\MM{l}+
(\MM{w}\cdot\nabla)(\MM{u}\cdot\nabla)\MM{l}\Bigg\rangle
+ \Bigg\langle {\frac{\delta \ell}{\delta a}}\diamond a,\MM{w}\Bigg\rangle
\\
& & \quad
+\Bigg\langle\phi,\mathcal{L}_{\MM{u}}\mathcal{L}_{\MM{w}}a
-\mathcal{L}_{\MM{w}}\mathcal{L}_{\MM{u}}a \Bigg\rangle
, \\
& =&\Bigg\langle \MM{\pi},-\left(\operatorname{ad}_{\MM{u}}\MM{w}\cdot\nabla\right)\MM{l}
\Bigg\rangle
+ \Bigg\langle \phi,\mathcal{L}_{\operatorname{ad}_{\MM{u}}\MM{w}}a
\Bigg\rangle + \Bigg\langle {\frac{\delta \ell}{\delta a}}\diamond a,\MM{w}\Bigg\rangle
| 2,858
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|
on}\gamma^\mu\lambda&=&
-\overline{\lambda}\gamma^\mu\epsilon&=&
-\left(\overline{\epsilon}\gamma^\mu\lambda\right)^\dagger\ ,\\
\overline{\epsilon}\gamma^\mu\gamma_5\lambda&=&
\overline{\lambda}\gamma^\mu\gamma_5\epsilon&=&
\left(\overline{\epsilon}\gamma^\mu\gamma_5\lambda\right)^\dagger\ ,\\
\overline{\epsilon}\gamma^\mu\gamma^\nu\lambda&=&
-\overline{\lambda}\gamma^\mu\gamma^\nu\epsilon&=&
\left(\overline{\epsilon}\gamma^\mu\gamma^\nu\lambda\right)^\dagger\ .
\end{array}$$ It is a useful exercise to establish any of these identities.
The Dirac Equation {#Sec3.5}
------------------
Let us now consider the dynamics of a single free Dirac spinor field, thus described, at the classical level, by complex valued Grassmann odd variables forming a 4-component Dirac spinor $\psi(x^\mu)$. The action principle for a such a system is given by the Lorentz invariant quantity $$S\left[\psi,\overline{\psi}\right]=\int d^4x^\mu\,
{\mathcal L}\left(\psi,\partial_\mu\psi\right)\ ,$$ with the Lagrangian density[^19] $${\mathcal L}=\frac{1}{2}i\left[\,\overline{\psi}\gamma^\mu\partial_\mu\psi\,-\,
\partial_\mu\overline{\psi}\gamma^\mu\psi\,\right]\,-\,m\overline{\psi}\psi\ .
\label{eq:DiracL}$$ Through the variational principle, the associated equation of motion is the celebrated Dirac equation, $$\left[i\gamma^\mu\partial_\mu\,-\,m\right]\,\psi(x^\mu)=0\ .
\label{eq:Dirac}$$
A few remarks are in order. Given the relations in (\[eq:chiral1\]) and (\[eq:chiral2\]), it is clear that the kinetic term $\overline{\psi}\gamma^\mu\partial_\mu\psi$ couples the chiral components of the Dirac spinor by preserving their chirality, while the coupling $m\overline{\psi}\psi$ switches between the two chirality components. As will become clear hereafter, since the real parameter $m\ge 0$ in fact determines the mass of the particle quanta associated to such a field, a massless Dirac particle propagates without flipping its chirality, whereas a massive particle sees both its chiral components contribute to its spacetime dynamics.
The term $m\o
| 2,859
| 3,186
| 3,158
| 2,761
| null | null |
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|
e consider as a dimension vector of $\Gamma$. Define $\varphihat'\in{\rm Rep}_{\Gamma,\v'}(\K)$ as the restriction of $\varphihat$ to $\Vhat'$. It is a non-zero subrepresentation of $\varphihat$. It is now possible to define a graded vector subspace $\Vhat''=\bigoplus_{i\in I}V_i''$ of $\Vhat$ such that the restriction $\varphihat''$ of $\varphihat$ to $\Vhat''$ satifies $\varphihat=\varphihat''\oplus\varphihat'$: we start by taking any subspace $V_{[i,j]}''$ such that $V_{[i,j]}=V_{[i,j]}'\oplus V_{[i,j]}''$, then define $V_{[i,j+r]}''$ from $V_{[i,j]}''$ as $V_{[i,j+r]}'$ was defined from $V_{[i,j]}$, and finally put $V''_i:=V_i$ if the vertex $i$ is not one of the vertices $[i,j],[i,j+1],\dots,[i,s_i]$. As $v_0> 0$, the subrepresentation $\varphihat''$ is non-zero, and so $\varphihat$ is not indecomposable.
We denote by ${\rm Rep}_{\Gamma,\v}^*(\F_q)$ be the subspace of representation $\varphihat\in{\rm Rep}_{\Gamma,\v}(\F_q)$ such that $\varphi_\gamma$ is injective for all $\gamma\in \Omega^0$, and by ${\rm M}_{\Gamma,\v}^*(\F_q)$ the set of isomorphism classes of ${\rm
Rep}^*_{\Gamma,\v}(\F_q)$. Put $M_{\Gamma,\v}^*(q)=\#\left\{{\rm
M}_{\Gamma,\v}^*(\F_q)\right\}$. Following [@crawley-par] we say that a dimension vector $\v$ of $\Gamma$ is *strict* if for each $i=1,\dots,k$ we have $n_0\geq v_{[i,1]}\geq
v_{[i,2]}\geq\cdots\geq v_{[i,s_i]}$. Let us denote by $\mathcal{S}$ the set of strict dimension vector of $\Gamma$.
$$\Log\left(\sum_{\v\in\mathcal{S}}M_{\Gamma,\v}^*(q)X^{\v}\right)
=\sum_{\v\in\mathcal{S}-\{0\}}A_{\Gamma,\v}(q)X^{\v}.$$ \[hua-inj\]
Let us denote by $I_{\Gamma,\v}(q)$ the number of isomorphism classes of indecomposable representations in ${\rm
Rep}_{\Gamma,\v}(\F_q)$. By the Krull-Schmidt theorem, a representation of $\Gamma$ decomposes as a direct sum of indecomposable representation in a unique way up to permutation of the summands. Notice that, for $\v\in\mathcal{S}$, each summand of an element of ${\rm Rep}_{\Gamma,\v}^*(\F_q)$ lives in some ${\rm
Rep}_{\Gamma,\w}^*(\F
| 2,860
| 1,637
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| 2,588
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|
{}, we get the following abstract error estimation:
Assume [**[I1]{}**]{}-[**[I3]{}**]{} hold, $C$ be a given constant in $(0,1)$ and $\alpha \ge \frac{(1+\delta)^2C_1}{4(1-C)^2}$. Let $u$ and $u_h$ be the solution to equations (\[eq:ellipticeq\]) and (\[eq:dg\]), respectively. Then $$\3bar u-u_h\3bar \lesssim C_A\bigg(1 + \frac{(1+C_1)(1+\alpha)}{C\alpha} \bigg)
\left(\sum_{K\in \mathcal{T}_h} h_K^{2s} \|u\|_{s+1,K}^2\right)^{1/2}.$$
Finally, we derive the $L^2$ error estimation by using the standard duality argument. Let $\delta=1$, that is, the bilinear form $A(\cdot,\cdot)$ is symmetric. Consider the following problem $$\begin{cases}
-\Delta \phi=u-u_h\qquad &\mbox{in }\Omega,\\
\phi=0 &\mbox{on }\partial\Omega.
\end{cases}$$ Here again, we assume that the domain $\Omega$ satisfies certain condition such that $\phi$ has $H^r$ regularity, with $r>3/2$. Let $\phi_h\in V_h$ be an approximation to $\phi$ such that they satisfy Assumption [**[I3]{}**]{}. Clearly $$\begin{aligned}
\|u-u_h\|^2 &= (-\Delta \phi,\, u-u_h) = \sum_{K\in \mathcal{T}_h}(\nabla \phi,\nabla (u-u_h))_K-\sum_{e\in\mathcal{E}_h}\langle\{\nabla \phi\},\, [u-u_h]\rangle_e \\
&= A(\phi, \, u-u_h) = A(\phi-\phi_h,\, u-u_h)\\
&\le \frac{1+\alpha}{\alpha} \3bar \phi-\phi_h \3bar\, \3bar u-u_h\3bar \\
&\le \frac{1+\alpha}{\alpha}C_A \left(\sum_{K\in \mathcal{T}_h} h_K^{2\min\{r-1,s\}} \|\phi\|_{\min\{r,s+1\},K}^2\right)^{1/2} \3bar u-u_h\3bar.
\end{aligned}$$ This gives the following theorem
Assume [**[I1]{}**]{}-[**[I3]{}**]{} hold, $\delta=1$, $C$ be a given constant in $(0,1)$, $\alpha \ge \frac{(1+\delta)^2C_1}{4(1-C)^2}$, and the elliptic equation (\[eq:ellipticeq\]) has $H^r$ regularity with $r>3/2$. Let $u$ and $u_h$ be the solution to equations (\[eq:ellipticeq\]) and (\[eq:dg\]), respectively. Then $$\|u-u_h\| \le \frac{1+\alpha}{\alpha}C_AC_R h^{\min\{r-1,s\}} \3bar u-u_h\3bar.$$
Requirements on meshes and discrete spaces
==========================================
On triangular or quadrilateral meshes, the usual tool for pr
| 2,861
| 1,148
| 1,686
| 2,810
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|
ongleftrightarrow}}}x
\text{ from $v$ such that }v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}{\underline{b}}\}.\end{aligned}$$ On the event $\{v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}\setminus
E_{{{\bf m}}+{{\bf n}}}(v,x;{{\cal A}})$, we take the *first* pivotal bond $b$ for $v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x$ from $v$ satisfying $v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}{\underline{b}}$. Then, we have (cf., [(\[eq:0th-ind-fact\])]{}) $$\begin{aligned}
{\label{eq:1st-ind-fact}}
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}$}}}={\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E_{{{\bf m}}+{{\bf n}}}(v,x;{{\cal A}})$}}}+\sum_{b\in
{{\mathbb B}}_\Lambda}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E_{{{\bf m}}+{{\bf n}}}(v,{\underline{b}};{{\cal A}})\text{ off }b\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b+n_b>0\}$}}}
\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\text{ in }{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(v){^{\rm c}}\}$}}}.\end{aligned}$$ Let $$\begin{aligned}
{\label{eq:Theta-def}}
\Theta_{v,x;{{\cal A}}}[X]=\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=v{\vartriangle}x}}\frac{
w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E_{{{\bf m}}+{{\bf n}}}(v,x;{{\cal A}})$}}}\,X({{\bf m}}+{{\bf n}}),&&
\Theta_{v,x;{{\cal A}}}=\Theta_{v,x;{{\cal A}}}[1].\end{aligned}$$ Substituting [(\[eq:1st-ind-fact\])]{} into [(\[eq:lmm-through\])]{}, we obtain (see Figure \[fig
| 2,862
| 1,108
| 1,403
| 2,832
| null | null |
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|
2}=4m_0^2$ with $n=n^{\prime}=0$.](prl1.eps){width="3in"}
It is easy to show that this function has a maximum near the threshold. If we consider $\omega$ near $2m_0$, the function $\mu_{\gamma}^{(2)}=f(X)$, where $X=\sqrt{4m_0^2-\omega^{2}}$ has a maximum for $X= {\pi\phi_{00}^{(2)}/m_0}^{1/3}$, which is very close to the threshold.
Thus, near the first threshold and in the second mode of propagation the expression (\[mm2\]) has a maximum value when $k_\perp^2 \simeq k_\perp^{\prime 2}$. Therefore in a vicinity of the first pair creation threshold the magnetic moment of the photon has a resonance peak which is positive, indicating a paramagnetic behavior, and its value is given by $$\mu_{\gamma}^{(2)}=\frac{m_0^2(B+2B_c)}{3m_\gamma
B^2}\left[2\alpha\frac{B}{B_c}\exp\left(-\frac{2B_c}{B}\right)\right]^{2/3}
\label{mumax}$$ Obviously, (\[mumax\]) would vanish also for $B\to 0$. The maximum of (\[mumax\]) is given numerically by $$\mu_\gamma^{(2)}\approx 3\mu^\prime
\left(\frac{1}{2\alpha}\right)^{1/3}\approx 12.85\mu^\prime$$
Thus, the maximum value achieved by the photon magnetic moment under the assumed conditions is larger than twice the anomalous magnetic moment of the electron.
In (\[mumax\]) we introduced the quantity $m_{\gamma}$ which has meaning near the thresholds, and which could be named as the “dynamical mass” of the photon in presence of a strong magnetic field, which is defined by the equation $$m_{\gamma}^{(2)}=\omega (k_\perp^{\prime
2})=\sqrt{4m_0^2-m_0^2\left[2\alpha
\frac{B}{B_c}\exp\left(-\frac{2B_c}{B}\right)\right]^{2/3}}
\label{dm}$$
The “dynamical mass” accounts for the fact that the massless photon coexists with the massive pair near the thresholds, leading to a behavior very similar to that of a neutral massive vector particle bearing a magnetic moment. However, it does not violate gauge invariance since the condition $\Pi_{\mu \nu}(0,0,B) =0$ is preserved. The idea of a photon mass has been introduced previously, for instance in ref.[@Osipov], in a regime different from ours
| 2,863
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| 2,844
| 3,853
| 0.769688
|
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|
nd eigenvectors determined here with a 30 state expansion for each $\theta$-parity. The ordering convention adopted for the states was taken as $$\notag \Psi^{+}_{0,-2}, \Psi^{+}_{0,-1},...\Psi^{+}_{5,2} ,
\Psi^{-}_{1,-2}...\Psi^{-}_{6,2}$$ yielding a Hamiltonian matrix blocked schematically into
$$\left( \begin{array} {cc}
H^{++}& H^{+-} \\
H^{-+} & H^{--} \end{array} \right)$$
Results
=======
Rather than present a large number of tables conveying little useful information per unit page length, the focus will be on indicating how some low-lying states evolve as a function of magnetic field strength for two distinct orientations. Some remarks will also be made regarding the general trend seen for higher excited states. Here the ratio $\alpha = a/R$ was set to $1/2$ as a compromise between smaller $\alpha$ where the states tend towards decoupled ring functions and larger $\alpha$ which are less likely to be physically realistic.
Fig. 1 illustrates the evolution of the energy eigenvalue for five low-lying states as a function of $\tau_0$ with $\tau_1 = 0$. The states are all distinct and are labelled in the caption. Not shown are values trivially obtained from the $\pm \nu B_0$ splitting arising from $B_0 = 0, \nu \neq 0$ degeneracy. It is interesting that level crossings with attendant movement towards a ground state with different $K\nu$ occurs near integer values of $\tau_0$, though it is not immediately clear if this is of real significance. It is also of interest to show the sensitivity of the dependence of $\Psi^*\Psi F$ on field strength. Fig. 2 shows that even for moderate field values ($\tau_0 = 5$ corresponds to a field of $1.3 \ T$ for a torus with $R = 50 \ nm$) the large effective flux as compared to atomic or molecular dimensions causes substantial modification to $\Psi^*\Psi F$ in the ground state.
The results given in Figs. 1-2 were for a field configuration that did not mix azimuthal basis states. To investigate an asymmetric case, let $\tau_0 = 0$ and vary $\tau_1$ wherein no field thr
| 2,864
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|
ether $\phi$ lifts to a homomorphism $$\psi: \: \pi_1\left( [T^4/{\mathbb Z}_2] \right) \: \longrightarrow \:
{\mathbb Z}_4.$$ First note $$\pi_1\left( [T^4/{\mathbb Z}_2] \right) \: = \: {\mathbb Z}_2 \rtimes
{\mathbb Z}^4,$$ where the nontrivial element in ${\mathbb Z}_2$ acts as multiplication by $-1$ on ${\mathbb Z}^4$. The homomorphism $\phi$ is the projection to ${\mathbb Z}_2$. The maximal 2-group quotient of ${\mathbb Z}_2 \rtimes {\mathbb Z}^4$ is ${\mathbb Z}_2 \times
({\mathbb Z}_2)^4$, so any homomorphism ${\mathbb Z}_2 \rtimes {\mathbb Z}^8 \rightarrow {\mathbb Z}_4$ will factor through ${\mathbb Z}_2 \times ({\mathbb Z}_2)^4$. But in the map ${\mathbb Z}_4 \rightarrow {\mathbb Z}_2$, the generator of ${\mathbb Z}_4$ maps onto the generator of ${\mathbb Z}_2$. Since ${\mathbb Z}_2 \times
({\mathbb Z}_2)^4$ does not contain any element of order 4, there is no map ${\mathbb Z}_2 \times ({\mathbb Z}_2)^4
\rightarrow {\mathbb Z}_4$ that lifts the projection onto the first factor. Therefore, the ${\mathbb Z}_2$ gerbe is nontrivial. More generally, if $[T^4/{\mathbb Z}_{2k}]$ is a ${\mathbb Z}_k$ gerbe over ${\mathbb Z}_2$, where the $Z_{2k}$ acts by first projecting to ${\mathbb Z}_2$, then it is nontrivial.
[^15]: For further examples of Calabi-Yau threefolds with this property, see [*e.g.*]{} [@cd]. Examples include ${\mathbb P}^7[2,2,2,2]$ and $({\mathbb P}^1)^4$ with a degree (2,2,2,2) hypersurface. For both, the restriction of an ambient hyperplane class to the Calabi-Yau defines a line bundle which is invariant but not equivariant.
[^16]: The dimension of this sheaf cohomology group can be determined from index theory, and applies to any stable irreducible rank 4 bundle ${\cal E}$ on a K3 surface.
[^17]: It is a standard result that the moduli in an irreducible rank $r$ vector bundle ${\cal E}$ on K3 with $c_1({\cal E}) = 0$, $c_2({\cal E}) = c_2(T K3)$ is encoded in $24r + 1 - r^2$ hypermultiplets, or $2(24r+1-r^2$ half-hypermultiplets. Here, $r=4$.
[^18]: For simplicity,
| 2,865
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|
====================================================================
To extend this method to more general equations with advected quantities is very simple: take the Lagrangian obtained from equation (\[inverse map sec\]) and add variables to represent higher-order derivatives. For the sake of brevity we shall compute one example, the incompressible Euler equations, and briefly discuss the implications for the circulation theorem.
Multisymplectic form of incompressible Euler equations
------------------------------------------------------
We start with the reduced Lagrangian $$\ell[\MM{u},p,\rho] = \int_{\Omega}\frac{1}{2}\rho u_iu_i +
p(1-\rho){\mathrm{d}}V(\MM{x}),$$ where $p$ is the pressure and $\rho$ is the relative density, and add dynamical constraints to form the Lagrangian: $$L = \frac{1}{2}\rho u_iu_i + p(1-\rho) +
\pi_k\left(l_{k,t}+u_il_{k,i}\right) +\phi\left(\rho_{,t}+(\rho
u_i)_{,i}\right).$$ This Lagrangian is already affine in the first-order derivatives, so the Euler-Lagrange equations are automatically multisymplectic in these variables: $$\begin{pmatrix}
0 & 0 & \pi_k\partial_i & 0 & -\rho\partial_i & 0 \\
0 & 0 & 0 & 0 & -\partial_t -u_i\partial_i & 0 \\
-\pi_k\partial_i & 0 & 0 & -\partial_t-u_i\partial_i & 0 & 0 \\
0 & 0 & \partial_t+u_i\partial_i & 0 & 0 & 0 \\
\rho\partial_i & \partial_t+u_i\partial_i & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
\end{pmatrix}
\begin{pmatrix}
u_i \\
\rho \\
l_k \\
\pi_k \\
\phi \\
p \\
\end{pmatrix}
= \nabla H,$$ where the quantity $$H = -\left(\frac{1}{2}\rho u_iu_i + p(1-\rho)\right)$$ is negative of the Hamiltonian density.
Circulation theorem for advected quantities
-------------------------------------------
The conservation law for particle-relabelling follows exactly as in Section \[inverse map EPDiff\], and we obtain equation (\[circulation\]) as before. The difference is that now the momentum formula (momentum map) is $$\MM{m} = {\frac{\partial \ell}{\partial \MM{u}}} = -\pi_k\nabla l_k -\phi\diamond a$$ and so one obtains $${\frac{d }{d t}}
| 2,866
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|
ay obligate the person in two tefillos, whereas crossing the
dateline in a westward direction (effectively stepping into the next
day without nightfall) does not require a new tefilloh. These two
views regarding tefilloh are expounded upon in Rav Betzalel Stern’s
Betzel Hachochmo, and Rav Yechezkel Roth’s Emek Hateshuva.
Q:
add connection string at runtime
Is it possible to insert a connection string into a web.config file at runtime, if it isn't there? Would this work?
The connection string is for a ASP.NET user login and create account database.
A:
How do you mean "insert?" If you mean can you manually edit web.config: yes, you can. You'd add an entry for connectionStrings (if there wasn't one) an then a child node for your specific connectionString.
If you mean can you do it in code: theoretically yes. However, it is normally bad practice, and a pain the rear.
Q:
How to set variable from select statement in SQL Server
I have a SELECT statement that returns 3 values, and I want get the data in those values. Can you help me please?
My code is :
declare @id int
declare @selected_name varchar(50)
declare @selected_age int
declare @selected_salary money
Select
name, age, salary
from
people
where
id = @id
set @selected_name = name
set @selected_age = age
set @selected_salary = salary
A:
You can do :
select @selected_name = name, @selected_age = age, @selected_salary = salary
from people
where id = @id;
Make sue this will need to have a single/unique entry, if that is not the case then you need top clause :
select top (1) @selected_name = name, @selected_age = age, @selected_salary = salary
from people
where id = @id;
| 2,867
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| 0.803665
|
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|
s}}}}$, with $\eta$ the viscosity of the solvent. Here $\left\langle \triangle r_{\textrm{cls}}^{2}(n)\right\rangle $ is the mean square displacement of the clusters after $n$ cycles, defined as $$\left\langle \triangle r_{\textrm{cls}}^{2}(n)\right\rangle =\dfrac{1}{N_{n_{c}}}{ \sum_{i=1}^{N_{n_{c}}}\triangle\mathbf{r}_{\textrm{cls},i}(n)\cdot\triangle\mathbf{r}_{\textrm{cls},i}(n)},$$ where $N_{n_{c}}$ is the number of clusters with $n_c$ colloids and $\triangle\mathbf{r}_{\textrm{cls},i}(n)$ is the center-of-mass displacement of a cluster with $n_c$ colloids after $n$ cycles. In addition, the time required for a cluster to diffuse over its diameter $\sigma_{\textrm{cls}}$ is the so-called Brownian time scale $\tau_{B}$, given by ${\tau_{B}=\sigma_{\textrm{cls}}^{2}/D_{\textrm{cls}}}$ with the assumption that the diameter of the spherical cluster ${\sigma_{\textrm{cls}}=\sqrt[3]{n_{c}}\sigma_2}$. Hence, we have $$\dfrac{n\tau}{\tau_{B}}\simeq\frac{\left\langle \triangle r_{\textrm{cls}}^{2}(n)\right\rangle }{6\sqrt[3]{n_{c}^{2}}\sigma_{2}^{2}}
\label{eqn:time}$$
From Eq. (\[eqn:time\]) we derive an MC simulation time of about $10-20\tau_B$, depending on the number of colloids in the cluster. As an example, for clusters composed of ten colloids with diameter of $154\ \textrm{nm}$ in water ($\eta=1\ \textrm{mPa\ s}$) and at room temperature, we obtain a Brownian time $\tau_{B}\sim0.85\ \textrm{s}$. Compared to the time scales of experiments that typically last tens of minutes [@Wittemann2008; @Ingmar2011], our MC simulation time scales are much smaller. However, the validity of a similar model for a binary mixture of single colloidal particles and droplets has been demonstrated by qualitative and quantitative agreement between experimental and simulation results [@Ingmar2011].
The simulations are performed in a cubic box with $N_c=250$ colloidal dumbbells with the packing fraction $\eta_c=0.01$ and $N_d=10-44$ droplets with packing fraction $\eta_d=0.1$. To initialize our simulation, we start by randomly di
| 2,868
| 3,986
| 3,818
| 2,899
| 3,105
| 0.774858
|
github_plus_top10pct_by_avg
|
thbf x},{\mathbf x}')=k({\boldsymbol{\mathrm{r}}})$ where ${\boldsymbol{\mathrm{r}}}={\mathbf x}-{\mathbf x}'$, so the covariance only depends on the distance between the input points. In that case we can also work with the spectral density, which is the Fourier transform of the stationary covariance function $$\label{eq:fourier}
S({\boldsymbol{\omega}}) =
\mathcal{F}[k]
= \int k({\boldsymbol{\mathrm{r}}})e^{-\text{i}{\boldsymbol{\omega}}^{\mathsf{T}}{\boldsymbol{\mathrm{r}}}}d{\boldsymbol{\mathrm{r}}},$$ where again ${\boldsymbol{\mathrm{r}}}={\mathbf x}-{\mathbf x}'$.
The perhaps most commonly used covariance function within the machine learning context [@Rasmussen2006] is the *squared exponential* (SE) covariance function $$\begin{aligned}
\label{eq:SE}
k_{\textrm{SE}}({\boldsymbol{\mathrm{r}}}) &= \sigma_f^2\exp\left[ -\frac{1}{2l^2}\|{\boldsymbol{\mathrm{r}}}\|_2^2 \right],\end{aligned}$$ which has the following spectral density $$\begin{aligned}
\label{eq:SES}
S_\textrm{SE}({\boldsymbol{\omega}})&=\sigma_f^2 (2\pi)^{d/2} l^d
\exp\left[
-\frac{l^2 \| {\boldsymbol{\omega}} \|_2^2}{2}
\right],\end{aligned}$$ where $d$ is the dimensionality of ${\mathbf x}$ (in our case $d=2$). The SE covariance function is characterized by the magnitude parameter $\sigma_f$ and the *length scale* $l$. The squared exponential covariance function is popular due to its simplicity and ease of implementation. It corresponds to a process whose sample paths are infinitely many times differentiable and thus the functions modeled by it are very smooth.
Another common family of covariance functions is given by the Matérn class
$$\begin{aligned}
k_{\textrm{Matern}}({\boldsymbol{\mathrm{r}}})&=
\sigma_f^2
\frac{2^{1-\nu}}{\Gamma(\nu)}
\left(
\frac{\sqrt{2\nu}\|{\boldsymbol{\mathrm{r}}}\|_2}{l}
\right)^\nu
K_\nu\left(
\frac{\sqrt{2\nu}\|{\boldsymbol{\mathrm{r}}}\|_2}{l}
\right),
\\
S_{\textrm{Matern}}({\boldsymbol{\omega}})&=
\sigma_f^2\frac{2^d \pi^{d/2}\Gamma(\nu+d/2)(2\nu)^\nu}{\Gamma(\nu)l^{2\nu}}
\left(
\frac{2
| 2,869
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| 1,900
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| null | null |
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|
ental representations is related to the existence of two SU(N)-invariant tensors, namely the Kronecker symbols ${\delta^\alpha}_\beta$ and ${\delta_\alpha}^\beta$ and the totally antisymmetric symbols $\epsilon^{\alpha_1\cdots\alpha_N}$ and $\epsilon_{\alpha_1\cdots\alpha_N}$, which themselves are directly connected to the defining properties of SU(N) matrices, namely the fact that they are unitary, $U^\dagger=U^{-1}$, and of unit determinant, ${\rm det}\,U=1$. Particularised to the SU(2) case, these simple properties may easily be checked. Indeed, using the transformation rules recalled above for co- and contra-variant indices under the SU(2) action, one has, for instance, $${\delta'_\alpha}^\beta={U_\alpha}^{\alpha_1}\,
{\delta_{\alpha_1}}^{\beta_1}\,{U^\dagger_{\beta_1}}^\beta=
{\delta_\alpha}^\beta\ ,$$ a result which readily follows from the unitarity property of SU(2) elements, $U^\dagger=U^{-1}$. Likewise for the $\epsilon_{\alpha\beta}$ tensor, for instance, $$\epsilon'_{\alpha\beta}=
{U_\alpha}^{\alpha_1}\,{U_\beta}^{\beta_1}\,\epsilon_{{\alpha_1}{\beta_1}}=
\epsilon_{\alpha\beta}\ ,$$ a result which follows from the unit determinant value, ${\rm det}\,U=1$.
In the general SU(N) case, these considerations imply that the $N$-dimensional contravariant representation $\overline{\underline{\bf N}}$, the complex conjugate of the $N$-dimensional covariant one $\underline{\bf N}$, is also equivalent to the totally antisymmetry representation obtained through the $(N-1)$-times totally antisymmetrised tensor product of the latter representation with itself, $$a_{\alpha_1\cdots\alpha_{N-1}}=\epsilon_{\alpha_1\cdots\alpha_{N-1}\beta}
a^\beta\ .$$ However, the SU(2) case is distinguished in this regard by the fact that this transformation also defines a unitary transformation on representation space. In other words, the relations $$a^\alpha=\epsilon^{\alpha\beta}\,a_\beta\ \ \ ,\ \ \
a_\alpha=\epsilon_{\alpha\beta}\,a^\beta\ ,$$ establish the unitary equivalence of the two 2-dimensional SU(2) representations $\und
| 2,870
| 2,795
| 2,726
| 2,731
| null | null |
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|
is $$\begin{aligned}
S=\int dt\left(-m\sqrt{V}\sqrt{-G_{tt}-G_{\rho\rho}\dot\rho^2}-\frac{qQ}{2\rho^2}\right)\;.\end{aligned}$$ The Hamiltonian is $$\begin{aligned}
H&=\frac{-m\sqrt{V}G_{tt}}{\sqrt{-G_{tt}-G_{\rho\rho}\dot\rho^2}}+\frac{qQ}{2\rho^2}\nonumber\\
&=\sqrt{-G_{tt}(G^{\rho\rho}\pi^2+m^2V)}+\frac{qQ}{2\rho^2}\end{aligned}$$ where in the second line we have written the Hamiltonian in terms of the canonical momentum $\pi(t)$. In the WKB approximation, the tunneling amplitude through classically forbidden regions is proportional to $$\begin{aligned}
\Gamma\sim\exp{(i\int_{\rho_-}^{\rho_+} \pi d\rho )}
\label{eq:wkbexp}\end{aligned}$$ where $$\begin{aligned}
\pi=\sqrt{-G_{\rho\rho}\left(G^{tt}\left(\epsilon_+-\frac{qQ}{2\rho^2}\right)^2-m^2V\right)},
\label{eq:wkbpi}\end{aligned}$$ and $\rho_\pm$ are the classical turning points. In classically forbidden regions the integral is complex, and the factor (\[eq:wkbexp\]) suppresses the tunneling rate.
The turning points are located at $\rho_-=0$ and $$\begin{aligned}
\rho_+=\frac{Q\rho_0\sqrt{q^2-m^2}}{\sqrt{q^2Q^2-4m^2\rho_0^4}}\end{aligned}$$ Since $Q< 2\rho_0^2$, the outer turning point is only finite (and therefore the pair production rate is only nonzero) if $$\begin{aligned}
q\gtrsim m \frac{2\rho_0^2}{Q}\approx m\;.\end{aligned}$$ If this inequality is satisfied, the WKB integral gives $$\begin{aligned}
\int_0^{\rho_+} \pi d\rho \approx i\sqrt{\rho_0^2-\frac{Q^2}{4\rho_0^2}}\left(-m+\frac{qQ}{2\rho_0^2} {\rm tanh}^{-1} \left(\frac{2\rho_0^2}{q Q}m\right)\right)\;.
\label{eq:intwkbexp}\end{aligned}$$
Let us first examine this exponent in the limit $Q/L^2\ll 1$. This is the limit in which the bubble is most stable against tunneling to a larger, expanding bubble. Expanding (\[eq:intwkbexp\]) in $Q/L^2$, we find $$\begin{aligned}
\Gamma\sim \exp\left({-\frac{\pi m Q (-m+q~{\rm tanh}^{-1}(m/q)}{m L}}\right)\sim \exp\left[{-\pi \left(\frac{Q}{L^2}\right) \left(\frac{m^2}{q^2}\right)(mL)}\right]
\label{eq:wkbrate}\end{aligned}$$ where in the last step we h
| 2,871
| 4,059
| 2,525
| 2,719
| null | null |
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|
s' movement inside the bus is part of pre-movement time) was large, the first person leaves the bus 37 seconds after initialization of the alarm system (*t* = 37*s*), see [Fig 4a](#pone.0201732.g004){ref-type="fig"}. Process of path selection during experiment 1 is presented in [Fig 4](#pone.0201732.g004){ref-type="fig"}. At the beginning, only four persons decided to leave the bus. Firstly, they did not know what to do, the participants just stayed in a group and tried to look around ([Fig 4b](#pone.0201732.g004){ref-type="fig"}). The decision on path selection (*t* = 55*s*) was taken in discussion with a second group of five persons, when participants decided to follow the evacuation signs. In [Fig 4c](#pone.0201732.g004){ref-type="fig"} the moment when the decision was made was captured, a student can be seen on the left (indicated with an arrow), pointing out the suggested direction of motion.
{#pone.0201732.g004}
It is worth noting, that *after selection of an evacuation path, all remaining participants followed*, without stopping (and discussing), as presented in [Fig 4d](#pone.0201732.g004){ref-type="fig"}. When a subsequent evacuee leaves and notices a group of participants going in some direction he/she follows them without hesitation. After the first two groups (9 persons) we observe only 2-3 cases when evacuees read the evacuation sign, however they never stop and always follow the group. This is another example of herding behavior during evacuation.
In experiment 4 we changed the bus stopping position in the tunnel and participants had
| 2,872
| 1,404
| 1,694
| 2,444
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|
}$ left translates of $A^2\cap N$. \[lem:slicing\] implies that $A^2\cap N$ is a $K^3$-approximate group, and so the existence of $P_1$ follows from \[cor:ruzsa\] and the existence of $P_2$ follows from \[cor:chang.ag\].
In the special case in which $\Bbbk={\mathbb{C}}$, an argument of Breuillard and Green shows that the coset nilprogression appearing in \[cor:bgt\] can be replaced with simply a nilprogression, as follows.
\[cor:bg\] Let $n\in{\mathbb{N}}$ and $K\ge1$. Suppose that $A\subset GL_n({\mathbb{C}})$ is a finite $K$-approximate group. Then there is a nilprogression $P_1\subset A^{e^{O(n^3)}\log^{O(n^2)}2K}$ of rank at most $e^{O(n^2)}\log^{O(n)}2K$ such that $A$ is contained in a union of at most $\exp(e^{O(n^2)}\log^{O(n)}2K)$ left translates of $P_1$, and a nilprogression $P_2\subset A^{{e^{O(n^3)}K^{3n+3}\log^{O(n^2)}2K}}$ of rank at most $e^{O(n^2)}K^3\log^{O(n)}2K$ such that $A$ is contained in a union of at most $K^{O_n(1)}$ left translates of $P_2$.
For the convenience of the reader we reproduce the Breuillard–Green argument giving \[cor:bg\]. The argument is facilitated by the following two general results about complex linear groups, in which we write ${\text{\textup{Upp}}}_n({\mathbb{C}})$ to mean the group of upper-triangular $n\times n$ complex matrices.
\[thm:malcev\] Let $n\in{\mathbb{N}}$, and suppose that $G<GL_n({\mathbb{C}})$ is a soluble subgroup. Then $G$ contains a normal subgroup $U$ of index at most $O_n(1)$ that is conjugate to a subgroup of ${\text{\textup{Upp}}}_n({\mathbb{C}})$.
\[prop:red.tf\] Let $n,s\in{\mathbb{N}}$, and let $N$ be an $s$-step nilpotent subgroup of ${\text{\textup{Upp}}}_n({\mathbb{C}})$. Then there is a torsion-free $s$-step nilpotent group $\Gamma$ such that $N$ embeds into ${\mathbb{R}}^n/{\mathbb{Z}}^n\times\Gamma$.
Although [@bg.sol Proposition 3.2] does not include the statement that $\Gamma$ has the same step as $N$, one can easily obtain this by replacing ${\mathbb{R}}^n/{\mathbb{Z}}^n\times\Gamma$ with $({\mathbb{R}}^n/{\mathbb{Z}}^n)N$.
| 2,873
| 2,005
| 1,942
| 2,765
| 2,159
| 0.782153
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ype 2 Equivalence.** If $(f_{k})$ are functions on $J$ that are integrable on every interior subinterval, then the following are equivalent statements.
\(a) For every interior subinterval $I$ of $J$ there is an integer $m_{I}\geq0$, and hence a smallest integer $m\geq0$, such that certain indefinite integrals $f_{k}^{(-m)}$ of the functions $f_{k}$ converge in the mean on $I$ to an integrable function $\Theta$ which, unlike in Type 1 Equivalence, need not itself be an indefinite integral of some function $f$.
\(b) $c_{k}(\varphi)=\int_{J}f_{k}\varphi\rightarrow c(\varphi)$ for every $\varphi\in\mathcal{C}_{0}^{\infty}(J)$.
Since we are now given that $\int_{I}f_{k}^{(-m)}(x)dx\rightarrow\int_{I}\Psi(x)dx$, it must also be true that $f_{k}^{(-m)}\varphi^{(m)}$ converges in the mean to $\Psi\varphi^{(m)}$ whence $$\int_{J}f_{k}\varphi=(-1)^{m}\int_{I}f_{k}^{(-m)}\varphi^{(m)}\longrightarrow(-1)^{m}\int_{I}\Psi\varphi^{(m)}\left(\neq(-1)^{m}\int_{I}f^{(-m)}\varphi^{(m)}\right).$$
The natural question that arises at this stage is then: What is the nature of the relation (not function any more) $\Psi(x)$? For this it is now stipulated, despite the non-equality in the equation above, that as in the mean $m$-integral convergence of $(f_{k})$ to a *function* $f$, $$\Theta(x):=\lim_{k\rightarrow\infty}\delta_{k}^{(-1)}(x)\overset{\textrm{def}}=\int_{-\infty}^{x}\delta(x^{\prime})dx^{\prime}\label{Eqn: delta1}$$
*defines* the non-functional relation (“generalized function”) $\delta(x)$ integrally as a solution of the integral equation (\[Eqn: delta1\]) of the first kind; hence formally[^9] $$\delta(x)=\frac{d\Theta}{dx}\label{Eqn: delta2}$$
***End Tutorial2***
The above tells us that the “delta function” is not a function but its indefinite integral is the piecewise continuous *function* $\Theta$ obtained as the mean (or pointwise) limit of a sequence of non-differentiable functions with the integral of $d\Theta_{k}(x)/dx$ being preserved for all $k\in\mathbb{Z}_{+}$. What then is the delta (and not its inte
| 2,874
| 3,571
| 3,364
| 2,675
| 3,939
| 0.769132
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| 2,875
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| 3,164
| 2,906
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| 0.826455
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\int\,d\eta\,d\eta^\dagger\,\eta^\dagger=0\ \ ,\ \
\int\,d\eta\,d\eta^\dagger\,\eta^\dagger\eta=1\ \ ,\ \$$ while the result for any linear combination of these $\eta$-monomials is given by the appropriate linear combination of the resulting integrations (the usual integral over Grassmann even variables being also linear for polynomials).
It turns out that the choice corresponding to the supersymmetric harmonic oscillator in (\[eq:SUSYL\]) is given by (one has, by construction of the supercovariant derivatives, $(DX)^\dagger=D^\dagger X$ for the real superfield $X$) $$S[X]=\int dt\,d\eta\,d\eta^\dagger\,\left[
-\frac{1}{8}m\omega^2\left(D^\dagger X\right)\left(DX\right)\,-\,
\frac{1}{4}m\omega^2X^2\right]\ .$$ Working out the superspace components of this expression, it reduces to $$\begin{array}{r l}
S[x,\theta,\theta^\dagger,f]=\int dt\,\Big\{&
\frac{1}{8}m\omega^2\left[f^2+\frac{4}{\omega^2}\dot{x}^2+
\frac{2i}{\omega}\left(\theta^\dagger\dot{\theta}+
\theta\dot{\theta}^\dagger\right)\right]\,-\,\\
& \\
&-\frac{1}{2}m\omega^2\left(fx+\theta^\dagger\theta\right)\Big\}\ .
\end{array}$$ Since no time derivatives of the highest superfield component $f(t)$ contribute to this action, this degree of freedom is indeed auxiliary with a purely algebraic equation of motion given by $$f(t)=2x(t)\ .$$ Upon reduction of this auxiliary degree of freedom, one recovers precisely the Lagrange function in (\[eq:SUSYL\]), up to a total derivative in time $d/dt(-im\omega\theta^\dagger\theta/4)$, while for the remaining dynamical degrees of freedom $x(t)$, $\theta(t)$ and $\theta^\dagger(t)$, the supersymmetry transformations (\[eq:SUSYvariation3\]) coincide then exactly with those in (\[eq:SUSYvariation2\]).
Having achieved the construction of the harmonic oscillator with a single supersymmetry generator ${\mathcal N}=1$ from these different but complementary points of view, one may wonder whether generalisations to types of potentials other than the quadratic one in $X^2$, to more general dynamics, and for a larger number $\m
| 2,876
| 1,571
| 2,492
| 2,638
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emitters
=============================================
{width="40.00000%"}
In addition to characterizing fully coherent radiation, CL polarimetry allows us to determine whether the measured radiation contains an unpolarized contribution such as in the case of incoherent luminescence from bulk or nanostructured materials. This is shown in Fig. \[Fig4\], where we compare azimuthally averaged zenithal cross cuts of the polarized ($S_{0}\times DOP$) and unpolarized ($S_{0}\times(1-DOP)$) emission intensities for single-crystal Au, Si and GaAs and compare them to calculations.
The emission from Au at $\lambda_0 = 850$ nm in Fig. \[Fig4\](a) is expected for coherent TR (see also Fig. S2 in the supplement) and hence fully polarized. The data indeed shows excellent agreement with a calculated TR emission distribution. In the case of GaAs in Fig. \[Fig4\](b), the emission is dominated by very bright incoherent radiative band-to-band recombination measured at $\lambda_{0} = 850$ nm. This luminescence is fully isotropic and unpolarized *inside* the material, but large differences between $s$- and $p$- Fresnel transmission coefficients for the semiconductor-vacuum interface partially polarize the emission as seen in the data. Figure \[Fig4\](b) shows that unpolarized light is indeed dominant. The weak polarized emission has a very different angular emission distribution, that agrees well with Fresnel calculations (see Fig. S3 in the supplement for more information). Lastly, Si is a material that displays weak luminescence that it is comparable to TR [@Brenny_JAP14]. Indeed, the polarized intensity for Si at $\lambda_{0} = 650$ nm shown in Fig. \[Fig4\](c) constitutes $\sim 31.7\%$ of the total emission, which is much more significant than for GaAs, although unpolarized emission remains the dominating contribution.
These examples show that angle-resolved polarimetry measurements provide quantitative and precise information about the origin of emission of different materials. This technique enables
| 2,877
| 651
| 3,408
| 2,972
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y. $CFT_2$ also belongs to this case.
The modular group is generated by the $T$ and $S$ transformations. The T-transformation leads to $$(\alpha,\beta)\rightarrow (\alpha,\beta+\alpha), \hs{3ex}
T=\left( \ba{cc}
1&0\\
1&1
\ea
\right).$$ The $S$-transformation leads to $$(\alpha,\beta)\rightarrow (-\beta,\alpha),\hs{3ex}
S=\left( \ba{cc}
0&-1\\
1&0
\ea
\right).$$ Note that the S-transformation exchanges the identifications along the two cycles, instead of the two coordinates.
Modular Invariance
------------------
We start with the torus $(\alpha,\beta)$ with two identifications, $$\label{torus1}
(\phi,t)\sim(\phi+2\pi,t)\sim(\phi+\alpha,t+\beta).$$ Consider the symmetry transformation of the theory, $$\phi\rightarrow f(\phi),\ \ \ t\rightarrow f'(\phi)^d t,$$ combining with $$t\rightarrow t+g(\phi),$$ where we have set $c=1$ for simplicity. Under such transformations, $$(\phi,t)\rightarrow(\phi',t''),\nn$$ and $$(\phi+2\pi,t)\rightarrow (f(\phi+2\pi),f'(\phi+2\pi)^dt+g(\phi+2\pi)).\nn$$ $$(\phi+\alpha,t+\beta)\rightarrow (f(\phi+\alpha),f'(\phi+\alpha)^d(t+\beta)+g(\phi+\alpha)).\nn$$
We would like to find the symmetry transformations which are consistent with the torus identification. For arbitrary point $(\phi',t'')$, there should be two identifications, $$(f(\phi),t'')\sim(f(\phi+2\pi),f'(\phi+2\pi)^dt+g(\phi+2\pi))\sim(f(\phi+\alpha),f'(\phi+\alpha)^d(t+\beta)+g(\phi+\alpha))$$ where t”=f’()\^dt+g(). If the identifications above are proper, $f(\phi+2\pi)-f(\phi)$ and $f(\phi+\alpha)-f(\phi)$ should not depend on $\phi'$, so $$f(\phi)=\lambda \phi+q$$ since $\phi$ is real. The constant shift $q$ of $\phi$ does not matter, and we can set it vanishing and have $$f(\phi)=\lambda\phi.$$ Then the identifications become $$(\phi',t'')\sim(\phi'+f(2\pi),t''+g(\phi+2\pi)-g(\phi))\sim(\phi'+f(\alpha),t''+\lambda^d\beta+g(\phi+\alpha)-g(\phi)).$$ Similarly, $g(\phi+2\pi)-g(\phi)$ should not depend on $\phi'$, so $$g(\phi)=g_0(\phi)+k\phi+p$$ where $g_0(\phi)\sim g_0(\phi+2\pi)$. Moreover, $\lambda^d\beta+g(\phi+\a
| 2,878
| 1,204
| 2,445
| 2,658
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e $$a^mb^n=(a^{p+m+n}b^m)^{-1}(a^{p+m+n}b^n).$$It is clear that $a^{p+m+n}b^m,a^{p+m+n}b^n \in S$.
\[twostraightl\] If $S=F_{D}\cup \widehat{F}\cup \widehat{\Lambda}_{I,p,d}\cup \Sigma_{p,d,P}$ is a left I-order in $\mathcal{B}$, then it is straight.
From corollaries \[twostraightr\] and \[twostraightl\], we have the main result in this section which is the third result in this article.
Let $S$ be a two-sided subsemigroup of $\mathcal{B}$. If $S$ is a left I-order in $\mathcal{B}$, then it is straight.
[111]{} A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Vol. 1, Mathematical Surveys 7, *American Math. Soc.* (1961). L. Descalco, N. Rǔskuc Subsemigroups of the bicyclic monoid. Internat. *J. Algebra Comput.* **15 (2005),** 37-57.
J. B. Fountain and Mario Petrich, Completely 0-simple semigroups of quotients, *Journal of Algebra* **101** (1986), 365-402.
N. Ghroda and Victoria Gould, Semigroups of inverse quotients, *Periodica Mathematica Hungarica*, to appear. N. Ghroda, Bisimple inverse $\omega$-semigroups of left I-quotients, [*ArXiv:1008.3241*]{}.
V. Gould, Bisimple inverse $\omega$-semigroup of left quotients, *Proc. London Math. Soc.* **52** (1986), 95-118. E. S. Lyapin, Canonical form of elements of an associative system given by defining relations, *Leningrad Gos. Ped. Inst. Uch. Zap.* **89** (1953), 45-54 \[Russian\].
---
author:
- 'Xiao Fang$^*$, Shige Peng$^\dagger$, Qi-Man Shao$^*$, Yongsheng Song$^\ddagger$'
date: '*The Chinese University of Hong Kong$^*$, Shandong University$^\dagger$, Chinese Academy of Sciences$^\ddagger$*'
title: Limit theorems with rate of convergence under sublinear expectations
---
[**Abstract:**]{} Under the sublinear expectation $\mathbbm{E}[\cdot]:=\sup_{\theta\in \Theta} E_\theta[\cdot]$ for a given set of linear expectations $\{E_\theta: \theta\in \Theta\}$, we establish a new law of large numbers and a new central limit theorem with rate of convergence. We present some interesting special cases and discuss a related statistical infere
| 2,879
| 615
| 3,083
| 2,890
| null | null |
github_plus_top10pct_by_avg
|
105 (50.5%)
Secondary eye
Total **75 (100%)**
1---diagnosed before the study period 48 (64.0%)
2---diagnosed after the first study eye 15 (20.0%)
3---both eyes diagnosed at same time 12 (16.0%)
######
Description of the lesion and symptoms at diagnosis.
Variable Study eye---*n* (%)
------------------------------------------ ---------------------
Lesion type
Total **208 (100%)**
Classic 83 (39.9%)
Minimally classic 23 (11.1%)
Occult 71 (34.1%)
Other shapes 12 (5.8%)
RAP 9 (4.3%)
IPCV 3 (1.4%)
Unknown 19 (9.1%)
Lesion size
Total **208 (100%)**
\<1 disk 38 (18.3%)
1-2 disks 90 (43.3%)
\>2 disks 42 (20.2%)
Unknown 38 (18.3%)
Subretinal neovascular membrane location
Total **208 (100%)**
Yuxtapapillar 3 (1.4%)
Extrafoveal 22 (10.6%)
Yuxtafoveal 78 (37.5%)
Subfoveal 104 (50.0%)
Subfoveal + Yuxtapapillar 1 (0.5%)
Presence of symptoms at diagnosis
Total **208 (100%)**
Yes 208 (100%)
Symptoms at diagnosis
Sudden and progressive loss of VA 136 (65.4%)
Centra
| 2,880
| 6,317
| 1,137
| 1,339
| null | null |
github_plus_top10pct_by_avg
|
ng a stronger iteration axiom (but not a larger large cardinal).
With the conclusion of [@T3] restored, [@T4], [@LT2], and [@T] are re-instated. We shall then proceed to improve the results of the two latter ones.
PFA$(S)[S]$ and the role of $\omega_1$
======================================
*PFA$(S)$* is the Proper Forcing Axiom (PFA) restricted to those posets that preserve the (Souslinity of the) coherent Souslin tree $S$.
*PFA$(S)[S]$ implies $\varphi$* is shorthand for *whenever one forces with a coherent Souslin tree $S$ over a model of PFA$(S)$, $\varphi$ holds.* *$\varphi$ holds in a model of form PFA$(S)[S]$* is shorthand for *there is a coherent Souslin tree $S$ and a model of PFA$(S)$ such that when one forces with $S$ over that model, $\varphi$ holds.*
For discussion of PFA$(S)[S]$, see [@D2], [@To], [@LT1], [@LT2], [@T4], [@T], [@FTT], [@T6].
The following results appear in [@LT2] and [@T], respectively.
\[thm:paracompactcopy\] There is a model of form ${\mathrm}{PFA}(S)[S]$ in which a locally compact, hereditarily normal space is hereditarily paracompact if and only if it does not include a perfect pre-image of ${\omega}_1$.
\[thm:paracompactcountablytight\] There is a model of form ${\mathrm}{PFA}(S)[S]$ in which a locally compact normal space is paracompact and countably tight if and only if its separable closed subspaces are Lindelöf and it does not include a perfect pre-image of ${\omega}_1$.
**** is the assertion that every first countable perfect pre-image of $\omega_1$ includes a copy of $\omega_1$.
${\mathrm}{PFA}(S)[S]$ implies ****.
$\mathbf{PPI}$ was originally proved from PFA in [@BDFN]. Using $\mathbf{PPI}$, we are able to weaken “perfect pre-image" to “copy" in the improved version of the first theorem, but provably cannot in the second theorem.
\[thm:paracompactcopyallmodels\] There is a model of form PFA$(S)[S]$ in which a locally compact, hereditarily normal space is hereditarily paracompact if and only if it does not include a copy of $\omega_1$.
There is a locally
| 2,881
| 2,307
| 3,219
| 2,886
| 1,874
| 0.784817
|
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|
this to a countable dense family of $f$, it follows that $\mu^{(n)} \to \gamma_{\mathbb{C}}$ weakly almost surely.
In the general case, since ${\left\vert G^{(n)} \right\vert} \to \infty$, each subsequence of $\mu^{(n)}$ has a subsequence $\mu^{(n_j)}$ for which, say, ${\left\vert G^{(n_j)} \right\vert} \ge j$, so that by the above argument $\mu^{(n_j)}$ converges to $\gamma_{\mathbb{C}}$ almost surely as $j \to
\infty$. It follows that $\mu^{(n)}$ converges to $\gamma_{\mathbb{C}}$ in probability.
The next proposition deals with the $G$-circulant analogue of the Gaussian Unitary Ensemble (GUE), which, up to a choice of normalization, is distributed as $2^{-1/2}(X + X^*)$, where $X$ is the complex Ginibre ensemble mentioned above. Equivalently, the diagonal entries of the GUE are standard real Gaussian random variables, the off-diagonal entries are standard complex Gaussian random variables, and the entries are independent except for the constraint that the matrix is Hermitian. It is worth noting explicitly that while each entry of the GUE has (complex) variance 1, the variance of a diagonal entry and the real part of an off-diagonal entry differ by a factor of $2$. (Again, the special case for classical circulant matrices was observed earlier in [@Meckes].)
\[T:GUE-eigenvalues\] Let $G$ be a finite abelian group and let $\{Y_a \mid a \in G \}$ be random variables which are independent except for the constraint $Y_{a^{-1}} = \overline{Y_a}$, and such that $$Y_a \sim \begin{cases} \gamma_{\mathbb{R}}& \text{ if } a^2 = 1,\\
\gamma_{\mathbb{C}}& \text{ if } a^2 \neq 1.
\end{cases}$$ Then the eigenvalues $\bigl\{ \lambda_\chi \mid \chi \in \widehat{G}
\bigr\}$ of $M$ given by are independent, standard real Gaussian random variables.
Let $\{Z_a \mid a \in G\}$ be independent, standard complex Gaussian random variables. Then $\{Y_a \mid a \in G\}$ are distributed as $\bigl\{ 2^{-1/2}\bigl(Z_a + \overline{Z_{a^{-1}}}\bigr) \mid a \in
G\bigr\}$. Thus the eigenvalues $\lambda_\chi$ of $M$ in the present p
| 2,882
| 3,505
| 2,777
| 2,447
| 4,136
| 0.767846
|
github_plus_top10pct_by_avg
|
s the closure of this core, which in this case of the topology being induced by the filterbase, is just the core itself. $A_{1}$ by its very definition, is a positively invariant set as any sequence of graphs converging to **$\textrm{Atr}(A_{1})$ must be eventually in $A_{1}$: the entire sequence therefore lies in $A_{1}$. Clearly, from Thm. A3.1 and its corollary, the attractor is a positively invariant compact set. A typical attractor is illustrated by the derived sets in the second column of Fig. \[Fig: DerSets\] which also illustrates that the set of functional relations are open in $\textrm{Multi}(X)$; specifically functional-nonfunctional correspondences are neutral-selfish related as in Fig. \[Fig: DerSets\], 3-2, with the attracting graphical limit of Eq. (\[Eqn: attractor\_adherence\]) forming the boundary of (finitely)many-to-one functions and the one-to-(finitely)many multifunctions.
Equation (\[Eqn: attractor\_adherence\]) is to be compared with the *image definition of an attractor* [@Stuart1996] where $f(A)$ denotes the range and not the graph of $f$. Then Eq. (\[Eqn: attractor\_adherence\]) can be used to define a sequence of points $x_{k}\in A_{n_{k}}$ and hence the subset $$\begin{aligned}
\omega(A) & \overset{\textrm{def}}= & \{ x\in X\!:(\exists n_{k}\in\mathbb{N})(n_{k}\rightarrow\infty)(\exists x_{k}\in A_{n_{k}})\textrm{ }(f^{n_{k}}(x_{k})\rightarrow x)\}\nonumber \\
& = & \{ x\in X\!:(\forall N\in\mathcal{N}_{x})(\forall A_{i}\in\mathcal{A})(N\bigcap A_{i}\neq\emptyset)\}\label{Eqn: Def: omega(A)}\end{aligned}$$
as the corresponding attractor of $A$ that satisfies an equation formally similar to (\[Eqn: attractor\_adherence\]) with the difference that the filter-base $\mathcal{A}$ is now in terms of the image $f(A)$ of $A$, which allows the adherence expression to take the particularly simple form $$\omega(A)=\bigcap_{i\in\mathbb{N}}\textrm{Cl}(f^{i}(A)).\label{Eqn: omega(A)_intersect}$$ The complimentary subset excluded from this definition of $\omega(A)$, as compared to $\textrm{Atr}(A
| 2,883
| 2,149
| 3,144
| 2,692
| 2,082
| 0.782936
|
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|
',\omega,E',E)\in S^2\times I^2\ |\ \omega'\cdot\omega-\mu_{11}(E',E)=0\},$$ then we would have (assuming that pertinent functions are Borel integrable) \[k-11\] (K\_[11]{})(x,,E)=\_[S’I’]{} \_[11]{}(x,’,,E’,E)(x,’,E’)d(’)dE’. This expression has the pleasant feature that the differential cross section $\underline{\sigma}_{11}(x,\omega',\omega,E',E)$ becomes a measurable function on space $G\times S^2\times I^2$.
Indeed, the key observation here is that in , the last line can be written as an integral, $$\begin{gathered}
\int_{0}^{2\pi}\psi(x,\gamma_{11}(E',E,\omega)(s),E')ds
=\frac{1}{\sqrt{1-\mu_{11}(E',E)^2}}\int_{\Gamma(E',E,\omega)} \psi(x,\cdot,E')d\ell, \\
=
\frac{1}{\sqrt{1-\mu_{11}(E',E)^2}}\int_{S'} \chi_{{\mathcal{M}}}(\omega',\omega,E',E)\psi(x,\omega',E')d\rho(\omega')\end{gathered}$$ where $\int_{\Gamma(E',E,\omega)} (\cdots)d\ell$ is the path integral along the curve $\Gamma(E',E,\omega)$. Hence we get the expression (\[k-11\]).
Finally, for the sake of completeness, let us mention that the Compton-Klein-Nishina photon$\to$ electron (i.e. $1\to 2$) cross section, with corresponding operator $K_{12}$, is given by the following formulas $$\sigma_{12}(x,\omega',\omega,E',E)
={}&
\hat\sigma_{12}(x,E',E)\chi_{12}(E,E')
\delta(\omega'\cdot\omega-\mu_{12}(E',E)), \\
\hat{\sigma}_{12}(x,E',E):={}& \hat{\sigma}_{11}(x,E',E'-E)\frac{(1+E')^2(1-\mu_{11}(E',E'-E))^2}{\mu_{12}(E',E)^3} \\
\mu_{12}(E',E):={}&\Big(1+\frac{1}{E'}\Big)\sqrt{\frac{E}{E+2}}, \\
\chi_{12}(E',E):={}&\chi_{{\mathbb{R}}_+}(E-E_0)\chi_{{\mathbb{R}}_+}\big(\frac{2{E'}^2}{1+2E'}-E\big)\chi_{{\mathbb{R}}_+}(E'-E),$$ where $(\omega', E')$ is the (direction, energy) of the incident photon, while $(\omega,E)$ is the (direction, kinetic-energy) of the (outgoing) recoil electron. If the scattering angle is written as $\theta_{12}$, then $\omega\cdot\omega'=\cos(\theta_{12})=\mu_{12}(E',E)$.
\[moller\] [*Electron-electron scattering - Møller*]{}. We denote the corresponding differential cross section by $\sigma_{22}(x,\omega',\
| 2,884
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t $R_{L_\lambda}^G(1)(x)=|P_\lambda|^{-1}\#\{g\in G\,|\, g^{-1}xg\in P_\lambda\}$, hence $R_{\mathfrak{l}_\lambda}^{\mathfrak{g}}(1)$ is the Lie algebra analogue of $R_{L_\lambda}^G(1)$ and the two functions take the same values on elements of same type.\[Rl=RL\]
We have $$\calF^\mathfrak{g}\left(Q_{\mathfrak{l}_\lambda}^\mathfrak{g}\right)=q^{\frac{1}{2}(n^2-\sum_i\lambda_i^2)}R_{\mathfrak{l}_\lambda}^\mathfrak{g}(1).$$ \[fourprop2\]
Consider the $\C$-linear map $R_{\mathfrak{l}_\lambda}^\mathfrak{g}:\C(\mathfrak{l}_\lambda)\rightarrow\C(\mathfrak{g})$ defined by
$$R_{\mathfrak{l}_\lambda}^\mathfrak{g}(f)(x)=|P_\lambda|^{-1}\sum_{\{g\in G\,|\, g^{-1}xg\in \mathfrak{p}_\lambda\}}f(\pi(g^{-1}xg))$$where $\pi:\mathfrak{p}_\lambda\rightarrow\mathfrak{l}_\lambda$ is the canonical projection. Then it is easy to see that $Q_{\mathfrak{l}_\lambda}^\mathfrak{g}=R_{\mathfrak{l}_\lambda}^\mathfrak{g}(1_0)$ where $1_0\in\C(\mathfrak{l}_\lambda)$ is the characteristic function of $0\in\mathfrak{l}_\lambda$, i.e., $1_0(x)=1$ if $x=0$ and $1_0(x)=0$ otherwise. The result follows from the easy fact that $\calF^{\mathfrak{l}_\lambda}(1_0)$ is the identity function $1$ on $\mathfrak{l}_\lambda$ and the fact (see Lehrer [@lehrer]) that $$\calF^\mathfrak{g}\circ R_{\mathfrak{l}_\lambda}^\mathfrak{g}=q^{\frac{1}{2}(n^2-\sum_i\lambda_i^2)}R_{\mathfrak{l}_\lambda}^\mathfrak{g}\circ\calF^{\mathfrak{l}_\lambda}.$$
For $x\in\mathfrak{g}$, denote by $1_x\in{\rm Fun}(\mathfrak{g})$ the characteristic function of $x$ that takes the value $1$ at $x$ and the value $0$ elsewhere. Note that $\calF^\mathfrak{g}(1_x)$ is the linear character $\mathfrak{g}\rightarrow\C$, $t\mapsto \Psi({\rm Tr}\,(xt))$ of the abelian group $(\mathfrak{g},+)$. Hence if $f:\mathfrak{g}\rightarrow\C$ is a function which takes integer values, then $\calF^\mathfrak{g}(f)$ is a character (not necessarily irreducible) of $(\mathfrak{g},+)$. Since the Green functions $Q_{\mathfrak{l}_\lambda}^\mathfrak{g}$ take integer values, by Proposition \[fourprop2\] the function
| 2,885
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|
--C80 114.1 (5)
C14---C19---H19 119.8 O2---C79---N2 125.1 (6)
C18---C19---H19 119.8 O2---C79---H79 117.4
C21---C20---C25 118.8 (4) N2---C79---H79 117.4
C21---C20---P6 123.1 (4) N2---C80---H80A 109.5
C25---C20---P6 117.8 (4) N2---C80---H80B 109.5
C22---C21---C20 120.7 (5) H80A---C80---H80B 109.5
C22---C21---H21 119.7 N2---C80---H80C 109.5
C20---C21---H21 119.7 H80A---C80---H80C 109.5
C23---C22---C21 119.7 (5) H80B---C80---H80C 109.5
C23---C22---H22 120.2 N2---C81---H81A 109.5
C21---C22---H22 120.2 N2---C81---H81B 109.5
C24---C23---C22 120.3 (5) H81A---C81---H81B 109.5
C24---C23---H23 119.9 N2---C81---H81C 109.5
C22---C23---H23 119.9 H81A---C81---H81C 109.5
C23---C24---C25 120.5 (5) H81B---C81---H81C 109.5
C23---C24---H24 119.8
------------------- ------------- ------------------- -----------
Introduction {#S1}
============
The defining feature of the canonical Wnt pathway is the stabilization of cytosolic β-catenin, which enters the nucleus and activates Wnt target genes by binding to transcription factors of the T-cell factor/lymphoid enhancing factor (TCF/LEF) family ([@R12]; [@R28]). In the absence of Wnt ligands, β-catenin is phosphorylated by a multi-protein complex that marks it for ubiquitination and degradation by the proteasome. This β-catenin degradation complex contains the adenomatous polyposis coli (APC) tumor suppressor, scaffold protein Axin, glycogen synthase kinase 3β (GSK3β), and casein kinase 1 (Ck1). The action of this complex is inhibited upon binding of Wnt to its receptors. Experiments performed in *Drosophila* ([@R49]), *Xenopus* ([@R42]) and mice ([@R37]) demonstrated that the low- density lipoprotein receptor-related protein 5 (LRP5)/LRP6 (termed *Arrow* in *Drosophila*) acts as a co-receptor for Wn
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3 - 5*m**2*p**3 - m*p**3 + 3*m wrt m?
-1620*p**3 - 18
What is the second derivative of 24*k**5*u**2 + 53*k**4*u**3 - 2301*k*u**3 wrt k?
480*k**3*u**2 + 636*k**2*u**3
What is the third derivative of -1271*f*n**4*s + f*n**3*s - f*n**3 + f*n**2*s - 5*f*n**2 - 3*n*s - 24 wrt n?
-30504*f*n*s + 6*f*s - 6*f
Find the second derivative of 5*p**2*u**3 - 3*p**2*u + 110*p**2 + 66*p*u**3 - 8*u**3 wrt p.
10*u**3 - 6*u + 220
Find the second derivative of -2*p*x**3 + 2*p*x**2 + 2*p*x - 347*p + 48*x**3 - 4*x**2 - 2 wrt x.
-12*p*x + 4*p + 288*x - 8
Differentiate 1544*f + 916 wrt f.
1544
Find the second derivative of -f**5*k - 8*f**3*k - 272*f**2*k + 55*f*k - 34*k wrt f.
-20*f**3*k - 48*f*k - 544*k
What is the first derivative of 2*d**4 - 374*d**3 + 2319 wrt d?
8*d**3 - 1122*d**2
Find the first derivative of -47*a**3*o**2 + 12*a**3 + 5*a**2*o**3 - 5*a wrt o.
-94*a**3*o + 15*a**2*o**2
What is the second derivative of -3*i**3 - 20*i**2 + 37*i?
-18*i - 40
Find the second derivative of -97*h**2*v**2 - 2*h**2*v + 6*h**2 - 8*h*v - 2*h - v**2 - 23*v wrt h.
-194*v**2 - 4*v + 12
What is the third derivative of 717*h**5*y + 3*h**5 + h**2*y - 298*h**2 + y wrt h?
43020*h**2*y + 180*h**2
What is the first derivative of 303*c**3*j**3 - c**3*j**2 + 117*c**3 wrt j?
909*c**3*j**2 - 2*c**3*j
What is the second derivative of -5*f**5 + 112*f**3 - 436*f wrt f?
-100*f**3 + 672*f
Find the third derivative of -2*p*x**5 - 2*p*x**4 - 2*p*x**3 - 6*p*x**2 + 5 wrt x.
-120*p*x**2 - 48*p*x - 12*p
What is the second derivative of 79*o**5 + 2*o**4 - 186*o + 3 wrt o?
1580*o**3 + 24*o**2
Differentiate 18474*n*x + 6762*x wrt n.
18474*x
Find the first derivative of b*j*y - 2*b*x*y**2 + 9*b*y**2 - 40*j**2*x**2 - 2*j**2 + j*x*y**2 + 7*x**2 wrt j.
b*y - 80*j*x**2 - 4*j + x*y**2
Find the first derivative of -19*x**2 + 21*x + 424 wrt x.
-38*x + 21
What is the third derivative of 315*g**2*p**3 + 2*g**2*p**2 + 2*g*p**6 - 21*p**2 - 2 wrt p?
1890*g**2 + 240*g*p**3
What is the second derivative of -3181*y**4 + 21403*y?
-38172*y**2
Differentiate -2*c*g - 271*c + 88*g wrt c.
-2*g
| 2,887
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u _{n}(dx)=f_{n}(x)dx,$ $$\limsup_{n}d_{k}(\mu ,\mu _{n})\times \theta ^{\rho _{h}+\varepsilon
}(n)<\infty \label{reg10}$$for some $\varepsilon >0.$ Then $\mu (dx)=f(x)dx$ with $f\in W^{q,p}.$
Moreover, for $\delta ,\varepsilon >0$ and $n_{\ast }\in {\mathbb{N}}$, let $$\begin{aligned}
A(\delta )& =\left\vert \mu \right\vert ({\mathbb{R}}^{d})\times 2^{l(\delta
)(1+\delta )(q+k+d/p_{\ast })}\quad
\mbox{with $l(\delta )=\min
\{l:2^{l\times \frac{\delta }{1+\delta }}\geq l\}$}, \label{reg12'} \\
B(\varepsilon )& =\sum_{l=1}^{\infty }\frac{l^{2(q+k+d/p_{\ast }+\varepsilon
)}}{2^{2\varepsilon l}}, \label{reg12''} \\
C_{h,n_{\ast }}(\varepsilon )& =\sup_{n\geq n_{\ast }}d_{k}(\mu ,\mu
_{n})\times \theta ^{\rho _{h}+\varepsilon }(n). \label{reg11}\end{aligned}$$Then, for every $\delta >0$ $$\left\Vert f\right\Vert _{q,p}\leq C_{\ast }(\Theta +A(\delta )\theta
(n_{\ast })^{\rho _{h}(1+\delta )}+B(\varepsilon )C_{h,n_{\ast
}}(\varepsilon )), \label{reg12}$$$C_{\ast }$ being the constant in (\[reg4\]) and $\rho _{h}$ being given in (\[reg5\]).
**Proof of Lemma \[REG\]**. We will produce a sequence of measures $%
\nu _{l}(dx)=g_{l}(x)dx,l\in {\mathbb{N}}$ such that $$\pi _{k,q,h,p}(\mu ,(\nu _{l})_{l})\leq \Theta +A(\delta )\theta (n_{\ast
})^{\rho _{h}(1+\delta )}+B(\varepsilon )C_{h,n_{\ast }}(\varepsilon
)<\infty .$$Then by Lemma \[lemma-inter\] one gets $\mu (dx)=f(x)dx$ with $f\in
W^{q,p} $ and (\[reg12\]) follows from (\[reg4\]). Let us stress that the $\nu _{l}$’s will be given by a suitable subsequence $\mu _{n(l)}$, $%
l\in {\mathbb{N}}$, from the $\mu _{n}$’s.
**Step 1**. We define $$n(l)=\min \{n:\theta (n)\geq \frac{2^{2hl}}{l^{2}}\}$$and we notice that $$\frac{1}{\Theta }\theta (n(l))\leq \theta (n(l)-1)<\frac{2^{2hl}}{l^{2}}\leq
\theta (n(l)). \label{reg13}$$Moreover we define $$l_{\ast }=\min \{l:\frac{2^{2hl}}{l^{2}}\geq \theta (n_{\ast })\}.$$Since$$\theta (n(l_{\ast }))\geq \frac{2^{2hl_{\ast }}}{l_{\ast }^{2}}\geq \theta
(n_{\ast })$$it follows that $n(l_{\ast })\geq n_{\ast }.$
We take no
| 2,888
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|
bf{w}}),$$ and hence $$D_{\mathbf{u}}D_{\mathbf{v}}h({\mathbf{w}}) \geq \min\{ D_{\mathbf{u}}^2 h({\mathbf{w}}), D_{\mathbf{v}}^2 h({\mathbf{w}})\}.$$
By continuity we may assume ${\mathbf{u}}, {\mathbf{v}}, {\mathbf{w}}\in \Lambda_{++}$. Then the polynomial $$\begin{aligned}
& g(x,y,z):=h(x{\mathbf{u}}+y{\mathbf{v}}+z{\mathbf{w}}) = h({\mathbf{w}})z^d+ \big(D_{\mathbf{u}}h({\mathbf{w}}) x + D_{\mathbf{v}}h({\mathbf{w}})y\big)z^{d-1}+ \\
&+ \left( D_{\mathbf{u}}D_{\mathbf{v}}h({\mathbf{w}}) xy + \frac 1 2 D_{\mathbf{u}}^2 h({\mathbf{w}}) x^2 + \frac 1 2 D_{\mathbf{v}}^2 h({\mathbf{w}})y^2\right)z^{d-2}+ \cdots\end{aligned}$$ is hyperbolic with hyperbolicity cone containing the positive orthant. By Theorem \[direct\] (1) so is $\partial^{d-2} g /\partial z^{d-2}$, and hence the polynomial $$2\frac {\partial^{d-2} g} {\partial z^{d-2}} \big( (1,0,0)+ t(0,0,1) \big) = D_{\mathbf{u}}^2 h({\mathbf{w}}) + 2D_{\mathbf{u}}D_{\mathbf{v}}h({\mathbf{w}})t + D_{\mathbf{v}}^2 h({\mathbf{w}})t^2$$ is real–rooted. Thus its discriminant is nonnegative, which yields the desired inequality.
\[righttrace\] There is a solution to Problem \[central\] such that all but at most one of the ${\mathbf{v}}_i$’s have trace either zero or $\epsilon$.
Moreover, if there is a solution to Problem \[central\] which satisfies the condition in Conjecture \[maxmax2\], then there is such a solution such that all but at most one of the ${\mathbf{v}}_i$’s have trace either zero or $\epsilon$.
Let ${\mathbf{v}}_1, \ldots, {\mathbf{v}}_m$ be a solution to Problem \[central\], and let $\rho$ be the maximal zero. Suppose $0<\tr({\mathbf{v}}_1), \tr({\mathbf{v}}_2) <\epsilon$. By Remark \[hypid\] $\rho {\mathbf{e}}+ {\mathbf{1}}$ is in the hyperbolicity cone $\Gamma_+$ of $h[{\mathbf{v}}_1, \ldots, {\mathbf{v}}_m]$. Since also $-{\mathbf{e}}_1, -{\mathbf{e}}_2 \in \Gamma_+$ we have ${\mathbf{w}}:= \rho{\mathbf{e}}+{\mathbf{1}}-{\mathbf{e}}_1-{\mathbf{e}}_2 \in \Gamma_+$, and hence ${\mathbf{w}}$ is in the (closed) hyperbolicity cone of $g=
| 2,889
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| 2,749
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iscretized only at the final stage is fundamentally different from the method employed in the original study by [@ld08] in which the optimization problem was solved in a fully discrete setting (the two approaches are referred to as “optimize-then-discretize” and “discretize-then-optimize”, respectively, cf. [@g03]). A practical advantage of the continuous (“optimize-then-discretize”) formulation used in the present work is that the expressions representing the sensitivity of the objective functional $\R$, i.e. the gradients $\nabla^{L_2}\R$ and $\nabla\R$, are independent of the specific discretization approach chosen to evaluate them. This should be contrasted with the discrete (“discretize-then-optimize”) formulation, where a change of the discretization method would require rederivation of the gradient expressions. In addition, the continuous formulation allows us to strictly enforce the regularity of maximizers required in problem \[pb:maxdEdt\_E\]. Finally and perhaps most importantly, the continuous formulation of the maximization problem makes it possible to obtain elegant closed-form solutions of the problem in the limit $\E_0 \rightarrow 0$, which is done in § \[sec:3D\_InstOpt\_E0to0\] below. These analytical solutions will then be used in §\[sec:3D\_InstOpt\_E\] to guide the computation of maximizing branches by numerically solving problem \[pb:maxdEdt\_E\] for a broad range of $\E_0$, as outlined in Algorithm \[alg:optimAlg\].
Extreme Vortex States in the Limit $\E_0 \to 0$ {#sec:3D_InstOpt_E0to0}
===============================================
It is possible to find analytic solutions to problem \[pb:maxdEdt\_E\] in the limit $\E_0 \to 0$ using perturbation methods. To simplify the notation, in this section we will drop the subscript $\E_0$ when referring to the optimal field. The Euler-Lagrange system representing the first-order optimality conditions in optimization problem \[pb:maxdEdt\_E\] is given by [@l69]
\[eq:KKT\_E\] $$\begin{aligned}
{\mathcal{B}}({\widetilde{\mathbf{u
| 2,890
| 1,815
| 1,056
| 2,775
| 4,091
| 0.768174
|
github_plus_top10pct_by_avg
|
al{G}^{1/3}}+
\frac{\mathcal{G}^{1/3}}{k_\perp^{\prime\prime
2}-k_{\perp}^{\prime 2}}\right] \label{fi}$$ where $\mathcal{G} =6 \pi\sqrt{3}D-\Lambda^{*3}+54 \pi^2
\phi_{n,n^{\prime}}^{(i) 2}(k_\perp^{\prime\prime
2}-k_\perp^{\prime 2})^2$ with $$D=\sqrt{-(k_\perp^{\prime 2}-k_\perp^{\prime\prime
2})^2\Lambda^{*3}\phi_{n,n^{\prime}}^{(i) 2}\left[1-\frac{27 \pi^2
\phi_{n,n^{\prime}}^{(i)2}(k_\perp^{\prime\prime2}-k_\perp^{\prime
2})^2}{\Lambda^{*3}}\right]}$$
Besides (\[fi\]), there are two other solutions of the above- mentioned cubic equation resulting from the substitution of (\[eg5\]) in (\[egg\]). These are complex solutions and are located in the second sheet of the complex plane of the variable $z_1=\omega^2-k_\parallel^2$ but they are not interesting to us in the present context.
Now the magnetic moment of the photon can be calculated by taking the implicit derivative $\partial\omega/\partial B$ in the dispersion equation. From (\[egg\]) and (\[eg5\]) it is obtained that
$$\mu_\gamma^{(i)}=\frac{\pi}{\omega(\vert
\Lambda\vert^3-4\pi\phi_{n,n^\prime}^{(i)}m_n
m_{n^\prime})}\left[\phi_{n,n^{\prime}}^{(i)}\left(A\frac{\partial
m_n}{\partial B}+Q\frac{\partial m_{n^\prime}}{\partial
B}\right)-\Lambda^2\frac{\partial
\phi_{n,n^{\prime}}^{(i)}}{\partial B}\right] \label{mm2}$$
with $
A= 4m_{n^\prime}[z_1+(m_n+m_{n^\prime})(3m_n+m_{n^\prime})] $ and $
Q= 4m_{n}[z_1+(m_n+m_{n^\prime})(m_n+3m_{n^\prime})] $.
In the vicinity of the first threshold $k_\perp^{\prime\prime
2}=0$, $k_\perp^{\prime 2}=4m_0^2$ and $\partial m_n/\partial B=0$ when $n=0$, therefore for the second mode the photon magnetic moment is given by
$$\mu_\gamma^{(2)}=\frac{\alpha
m_0^3\left(4m_0^2+z_1\right)\exp\left(-\frac{k_{\perp}^2}{2eB}\right)}{\omega
B_c\left[(4m_0^2+z_1)^{3/2}+\alpha m_0^3 \frac{B}{B_c}
\exp\left(-\frac{k_{\perp}^2}{2eB}\right)\right]}\left(1+\frac{k_{\perp}^2}{2eB}\right).\label{FRR1}$$
![\[fig:Phvk\] Photon magnetic moment curve drawn with regard to perpendicular momentum squared, for the second mode $k_\perp^{\prime
| 2,891
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|
ince $3\ls v\ls a-1$, we can take $\delta=\sigma$.
If $a\equiv0$, take $\gamma={\hat\Theta_{A}}+{\hat\Theta_{B}}$. By Proposition \[abhoms\], $\gamma$ is a homomorphism from $S^\mu$ to $S^\la$. By Lemma \[uab\] and Lemma \[countu\], $$\delta\circ\gamma=\mbinom{u-v}{a-v}\mbinom{u-a}2{\hat\Theta_{D}}.$$ The first term is odd by assumption; the second term is odd because $u-a\equiv2$, and ${\hat\Theta_{D}}\neq0$ by Lemma \[uab\].
If $a\equiv2$, take $\gamma={\hat\Theta_{A}}$. Then $\gamma$ is a homomorphism from $S^\mu$ to $S^\la$, and $$\delta\circ\gamma=\mbinom{u-v}{a-v}\mbinom{u+v-a-1}2{\hat\Theta_{D}}.$$ Again, the first term is odd by assumption, the second term is odd because now $u+v-a-1\equiv2$, and ${\hat\Theta_{D}}\neq0$.
Conversely, suppose we have homomorphisms $\gamma,\delta$ such that $\delta\circ\gamma\neq0$. By Proposition \[cdhomdim1\] we can assume that $3\ls v\ls a-1$ and take $\delta=\sigma$. From Proposition \[muladim\] we can take $\gamma$ to be ${\hat\Theta_{A}}$, ${\hat\Theta_{B}}$ or ${\hat\Theta_{A}}+{\hat\Theta_{B}}$, according to the congruences in Proposition \[abhoms\]. Then $\delta\circ\gamma$ will be a scalar multiple of ${\hat\Theta_{D}}$, and the scalar will include $\mbinom{u-v}{u-a}$ as a factor. So this binomial coefficient must be odd, and all that remains is to show that $v\equiv3\ppmod4$. We consider the three cases of Proposition \[abhoms\]. Note that because $v>1$, Theorem \[irrspecht\] gives $u-v\equiv3$.
: In this case the coefficient of ${\hat\Theta_{D}}$ in $\delta\circ\gamma$ is $$\mbinom{u-v}{a-v}\mbinom{u+v-a-1}2.$$ The second binomial coefficient must be odd, so $u+v-a\equiv3$. In the case $a-v\equiv3$, this is the same as saying $u\equiv2$, so that $v\equiv3$. In the case $v=b+3$, we have $a=u$, so that again $v\equiv3$.
: Now the coefficient of ${\hat\Theta_{D}}$ in $\delta\circ\gamma$ is $$\mbinom{u-v}{a-v}\mbinom{u-a}2.$$ The second binomial coefficient is odd only if $u\equiv a+2$, which is the same as saying $v\equiv 3$.
: In this case the coefficie
| 2,892
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| 3,873
| 0.769612
|
github_plus_top10pct_by_avg
|
M_\pi(C^{\lambda},k)=\cM_\pi(C^{\lambda+\varepsilon},k)$. We claim one can find a projective curve $\cD\subset \PP^2_w$, $w=(w_0,w_1,w_2)$ as in in the proof of Lemma \[lemma:global-realization\], where $\deg_w\cD=c(1+\Delta w_2)$ for $c>0$, $c\equiv [C] \mod (w_2)$ and a big enough $\Delta\gg 0$ and $a\equiv k \mod (w_2)$ such that $\lambda':=\frac{a}{\deg_w\cD}\in [\lambda,\lambda+\varepsilon)$. Since $\cD$ is normal crossing outside of $P$ by property \[prop:3b\] in Lemma \[lemma:global-realization\] the morphism $\pi$ is also a resolution of $\cD$. Hence $$\cM_\pi(C^\lambda,k)=\cM_\pi(C^{\lambda'},k)=\cM_\pi(\cD^{\lambda'},k)=\cM_\pi(\cD^{\lambda'},a).$$ The map $\pi^{(a)}$ in Theorem \[thm:conucleo\_singular\] is as simple as $$\pi^{(a)}: H^0\left(\PP^2_w,\mathcal{O}_{\PP^2_w}\left( aH+K_{\PP^2_w} - \mathcal{D}^{(a)}\right) \right)
\longrightarrow
\frac{\mathcal{O}_{\PP^2_w,P}\left( aH+K_{\PP^2_w} - \mathcal{D}^{(a)}\right)}{\cM_\pi(\cD^{\lambda'},a)}.$$ Since $\deg_w\cD$ can be chosen big enough, one can assume $\pi^{(a)}$ is surjective. Since $\ker \pi^{(a)}=H^2(Y,\cO_Y(L^{(a)}))$ is a birational invariant of the associated covering of $\PP^2_w$ ramified along $\cD$, one obtains $$\dim_\CC \frac{\mathcal{O}_{\PP^2_w,P}\left( aH+K_{\PP^2_w} - \mathcal{D}^{(a)}\right)}{\cM_\pi(\cD^{\lambda'},a)}
=\dim_\CC \frac{\mathcal{O}_{\PP^2_w,P}\left( kH+K_{\PP^2_w} - \mathcal{D}^{(k)}\right)}{\cM_\pi(\cD^{\lambda'},k)}$$ is independent of the resolution.
Finally, the result follows since any two resolutions are simultaneously dominated by a third one and if $\pi$ dominates $\pi'$, then $\cM_{\pi}(C^\lambda,k)\subset \cM_{\pi'}(C^\lambda,k)$.
As for the claim, note that the proof of Lemma \[lemma:global-realization\] can be redone from the local situation using any list of positive $c_i\equiv [C_i] \mod (w_2)$ and $\deg_wD_i=c_i(1+\Delta w_2)$. Hence $\deg_wD=c:=\sum_i n_ic_i\equiv [C] \mod (w_2)$. Take any $\Delta\gg 0$ such that $\Delta\equiv -w_2^{-1} \mod (w_0w_1)$ and $\frac{w_2}{c(1+\Delta w_2)}<\varep
| 2,893
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| null | null |
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|
re indices $A\neq B$, which means that real wave vectors are excluded from our consideration (we do not consider here the mixed case for wave vectors involving real and complex wave vectors). Under the above hypotheses, the $k$ wave vectors (\[eq:wv\]) and their complex conjugates $$\bar{{\lambda}}^A(u)={\left( \bar{{\lambda}}_1^A(u),\ldots,\bar{{\lambda}}_p^A(u) \right)}\in{\mathbb{C}}^p,\quad A=1,\ldots,k,$$satisfy the dispersion relation (\[eq:2.3\]). In what follows it is useful to introduce the notation $c.c.$ which means the complex conjugate of the previous term or equation. This notation is convenient for computational purposes allowing the presentation of some expressions in abbreviated form.
####
Let us suppose that there exists a unique solution $u(x)$ of the system (\[eq:3.1\]) of the form
\[eq:3.5\] u=f(r\^1(x,u),…, r\^k(x,u),|[r]{}\^1(x,u),…,|[r]{}\^k(x,u))+c.c.,
where the complex-valued functions $r^A,\bar{r}^A:{\mathbb{R}}^p\times {\mathbb{R}}^q{\rightarrow}{\mathbb{C}}$ are called the Riemann invariants associated respectively to wave vectors $\lambda^A$, $\bar{\lambda}^A$ and are defined by
\[eq:3.2\] r\^A(x,u)=\_i\^A(u)x\^i,|[r]{}\^A(x,u)=|\_i\^A(u)x\^i,A=1,…,k,
where $c.c.$ means complex conjugated previous term. Note that the functions $u(x)$ are defined implicitly in terms of $u^\alpha, x^i$, $r^A$ and $\bar{r}^A$. For any function $f:{\mathbb{C}}^k{\rightarrow}{\mathbb{C}}^q$ and its complex conjugate, the equation (\[eq:3.5\]) determines a unique real-valued function $u(x)$ on a neighborhood of the origin $x=0$. Note also that an analogue analysis as presented below can be performed if one replace the postulated form of the solution (\[eq:3.5\]) written in the Riemann invariants by the expression $$u=i{\left( f(r^1(x,u),\ldots,r^k(x,u),\bar{r}^1(x,u),\ldots,\bar{r}^k(x,u))-c.c. \right)}.$$Therefore we omit this case.
####
The Jacobi matrix of derivatives of $u(x)$ is given by
\[eq:3.6\] u=(u\^\_i)=[( I\_q- )]{}\^[-1]{} [( [( + )]{}+c.c. )]{},
or equivalently as
\[eq:3.7\
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\tilde L^{n-4k}$ is obtained by gluing the manifold $\tilde L^{n-4k}_{x} \cup \tilde L^{n-4k}_y$ with the manifold $\tilde L^{n-4k}_z$ along the common boundary $\tilde
\Lambda^{n-4k-1}$. Note that the group of the framing of the last manifold $\tilde \Lambda^{n-4k-1}_z$ is the subgroup $\I_3 \subset
\Z/2 \int \D_4$.
Let $OP\alpha$ be the $\Z/2 \int \D_4$–framed immersion obtained from an arbitrary $\Z/2 \int \D_4$-framed immersion $\alpha$ by changing the structure group of the framing by the transformation $OP$. The $\Z/2 \int \D_4$-framed manifold (with boundary) $(\tilde L^{n-4k}_y, \tilde \Psi_y, \tilde \zeta_y)$ coincides with the two disjoint copies of $\Z/2 \int \D_4$-framed manifold (with boundary) $OP(\tilde L^{n-4k}_y, \tilde \Psi_y,
\tilde \zeta_y)$.
Let us put $\alpha_1=-OP(\tilde L^{n-4k}, \tilde \Psi, \tilde
\zeta)$. Let us define the sequence of $\Z/2 \int \D_4$-framed immersions $\alpha_2 = -2 OP\alpha_1$, $\alpha_3 = -2 OP\alpha_2$, $\dots$, $\alpha_j = -2 OP\alpha_{j-1}$.
Obviously, the $\D/4 \int \Z/2$-framed immersion $\alpha_1 +
\alpha_2 = \alpha_1 + 2OP\alpha_1^{-1}$ is represented by 3 copies of the manifold $\tilde L^{n-4k}$. The second and the third copies are obtained from the first copy by the mirror image and the changing of structure group of the framing. The manifold $-OP[\tilde L^{n-4k}] \cup 2[\tilde L^{n-4k}]$ contains, in particular, a copy of $-OP[\tilde L^{n-4k}_x]$ inside the first component and the union $[\tilde L^{n-4k}_y \cup L^{n-4k}_y]$ of the mirror two copies of $-OP[\tilde L^{n-4k}_x]$ in the second and the third component. Therefore the manifold $-OP[\tilde
L^{n-4k}] \cup 2[ \tilde L^{n-4k}]$ is $\Z/2 \int \D_4$-framed cobordant to a $\Z/2 \int \D_4$-framed manifold, obtained by gluing the union of a copy of $-OP[\tilde L^{n-4k}_x]$ and 4 copies of $\tilde L^{n-4k}_y$ by a $\I_3$-framing manifold along the boundary. This cobordism is relative with respect to the submanifold $-OP[\tilde L^{n-4k}_z] \cup 2[L^{n-4k}_z] \subset
-OP[L^{n-4k}] \cup 2L^{n-4k}$.
By an
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@MR2002d:14084]; we reproduce the list of curves obtained in [@MR2002d:14084] in an appendix at the end of this paper (§\[appendix\]). For another classification, from a somewhat different viewpoint, we refer to [@MR1698902]. For these curves, the limits can be determined using the results in [@MR2002d:14083] (see also §\[boundary\]). The following statement reduces the computation of the limits of an arbitrary curve ${{\mathscr C}}$ to the case of curves with small orbit.
\[main\] Let ${{\mathscr X}}$ be a limit of a plane curve ${{\mathscr C}}$ of degree $d$, obtained by applying to it a ${{\mathbb{C}}}((t))$-valued point of ${\text{\rm PGL}}(3)$ with singular center. Then ${{\mathscr X}}$ is in the orbit closure of a star (reproducing projectively the $d$-tuple cut out on ${{\mathscr C}}$ by a line meeting it properly), or of curves with small orbit determined by the following features of ${{\mathscr C}}$:
- The linear components of the support ${{{{\mathscr C}}'}}$ of ${{\mathscr C}}$;
- The nonlinear components of ${{{{\mathscr C}}'}}$;
- The points at which the tangent cone of ${{\mathscr C}}$ is supported on at least $3$ lines;
- The Newton polygons of ${{\mathscr C}}$ at the singularities and inflection points of ${{{{\mathscr C}}'}}$;
- The Puiseux expansions of formal branches of ${{\mathscr C}}$ at the singularities of ${{{{\mathscr C}}'}}$.
The limits corresponding to these features may be described as follows. In cases I and III they are unions of a star and a general line, that we call ‘fans’; in case II, they are supported on the union of a nonsingular conic and a tangent line; in case IV, they are supported on the union of the coordinate triangle and several curves from a pencil $y^c=\rho\, x^{c-b} z^b$, with $b<c$ coprime positive integers; and in case V they are supported on unions of quadritangent conics and the distinguished tangent line. The following picture illustrates the limits in cases IV and V:
A more precise description of the
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_{m,m^\prime} \delta_{h,h^\prime} \delta_{k,k^\prime}
\,.
\label{eq:orthogonal-relation}$$ Here the overbar denotes complex conjugation, and the volume element is given by $$\int_{\Sigma_u} {\mathrm{dVol}\,}= \lim_{T\rightarrow \infty}
\int_{-T}^T\text{d}\tau \int_0^{2\pi}\text{d}\varphi
\int_0^{\pi}\text{d}\psi \sqrt{-\gamma}
\,,$$ where $\gamma$ is the determinant of the three-dimensional metric, and in these coordinates $\sqrt{-\gamma} =
2\csc^{2}\psi\sqrt{1-u^{4}}$. To prove Eq. we first note the basis components ${v}_j^{(m\,h\,k)}$ in global coordinates have the $\tau$ and $\varphi$ dependence, $${v}_j^{(m\,h\,k)} \sim \exp{(i m\varphi)\exp{\left[i(h-k)\tau\right]}}.
\label{eq:vector-basis-general-form}$$ This dependence on $\tau$ and $\varphi$ is the same for the scalar and tensor basis components. Once we integrate over $\varphi$ and $\tau$ in Eq. , the integral will be proportional to $\delta_{m,m^\prime}\delta_{h-k,h^\prime-k^\prime}$. Notice that the boundaries $\tau\rightarrow \pm \infty$ are oscillatory, so the $\tau$ integral needs to be regulated in the same way as Fourier integrals.
Now we only need to show bases with different weight $k$ are orthogonal. Once this is done we will recover Eq. . For simplicity, from now on we only track the $k$-index in the vector bases. Recall that we obtain the lower weight bases by applying the lowering operator order by order, $$\begin{aligned}
\langle {\bf u}^{(k)}, {\bf v}^{(k^\prime)} \rangle
= \langle {\bf u}^{(k)}, \mathcal{L}_{L_-}{\bf v}^{(k^\prime-1)}
\rangle
\,.\end{aligned}$$ Now we try to “integrate by parts” with the Lie derivative,
$$\begin{aligned}
\label{eq:boundary-extraction}
\langle {\bf u}^{(k)}, \mathcal{L}_{L_-}{\bf v}^{(k^\prime-1)}
\rangle
&= \int_{\Sigma_u} \mathcal{L}_{L_-}\left(\overline{u_i^{(k)}} v^i_{(k^\prime)}\right) {\mathrm{dVol}\,}- \langle \overline{\mathcal{L}_{L_-}}{\bf u}^{(k)}, {\bf v}^{(k^\prime-1)} \rangle, \\
&= \int_{\Sigma_u} \mathcal{L}_{L_-}\left(\overline{u_i^{(k)}} v^i_{(k^\prime)}\right) {\mathrm{dVol
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pchi}$$ for all $i,j\in I\setminus \{p\}$. It is a small exercise to check that then $({\sigma }_p^\chi )^*\chi $ is $p$-finite, and $$\begin{aligned}
\label{eq:rp2}
c_{pj}^{r_p(\chi )}=c_{pj}^\chi \quad \text{for all $j\in I$},
\qquad r_p^2(\chi )=\chi .\end{aligned}$$ The reflections $r_p$, $p\in I$, generate a subgroup $$\begin{aligned}
{\mathcal{G}}=\langle r_p\,|\, p\in I\rangle\end{aligned}$$ of the group of bijections of the set ${\mathcal{X}}$. For all $\chi \in {\mathcal{X}}$ let ${\mathcal{G}}(\chi )$ denote the ${\mathcal{G}}$-orbit of $\chi $ under the action of ${\mathcal{G}}$.
Let $\chi \in {\mathcal{X}}$ such that $\chi '$ is $p$-finite for all $\chi '\in {\mathcal{G}}(\chi )$ and $p\in I$. By Eq. we obtain that $${\mathcal{C}}(\chi )=
{\mathcal{C}}(I,{\mathcal{G}}(\chi ),(r_p)_{p\in I},
(C^{\chi '})_{\chi '\in {\mathcal{G}}(\chi )})$$ is a connected Cartan scheme. The Weyl groupoid of $\chi $ is then the Weyl groupoid of the Cartan scheme ${\mathcal{C}}(\chi )$ and is denoted by ${\mathcal{W}}(\chi )$. Clearly, ${\mathcal{C}}(\chi )={\mathcal{C}}(\chi ')$ and ${\mathcal{W}}(\chi )={\mathcal{W}}(\chi ')$ for all $\chi '\in {\mathcal{G}}(\chi )$.
\[ex:Cartan\] Let $C=(c_{i j})_{i,j\in I}$ be a generalized Cartan matrix. Let $\chi \in {\mathcal{X}}$, $q_{ij}=\chi ({\alpha }_i,{\alpha }_j)$ for all $i,j\in I$, and assume that $q_{ii}^{c_{ij}}=q_{ij}q_{ji}$ for all $i,j\in I$, and that $\qnum{m+1}{q_{ii}}\not=0$ for all $i\in I$ and $m\in {\mathbb{N}}_0$ with $m<\max \{-c_{ij}\,|\,j\in I\setminus \{i\}\}$. (The latter is not an essential assumption, since if it fails, then one can replace $C$ by another generalized Cartan matrix $\tilde{C}$, such that $\chi $ has this property with respect to $\tilde{C}$.) One says that $\chi $ is of *Cartan type*. Then $\chi $ is $i$-finite for all $i\in I$, and $c_{ij}^\chi =c_{ij}$ for all $i,j\in I$. Eq. gives that $$\begin{aligned}
r_p(\chi )({\alpha }_i,{\alpha }_i)=&q_{ii}=\chi ({\alpha }_i,{\alpha }_i),\\
r_p(\chi )({\alpha }_i,{\alpha }_j)\
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${\prod{\Phi}} = (\thinspace{\prod{\Psi}}) \mid R$. By Theorem \[T:CHC\_RSTR\_EQ\_RSTR\_CHC\], the restriction of the choice space equals the choice space of the restriction: $(\thinspace\prod\Psi) \mid R = \prod ({{\Psi}\negmedspace\mid\negmedspace{R}})$. Transitivity of equality implies ${\prod{\Phi}} = \prod ({{\Psi}\negmedspace\mid\negmedspace{R}})$. Then, by Theorem \[T:SPACE\_UNIQ\_ENSEMBLE\] (invertibility of the Cartesian product), $\Phi = {{\Psi}\negmedspace\mid\negmedspace{R}}$.
Suppose term $(i,P) \in \Phi$. Since $\Phi = {{\Psi}\negmedspace\mid\negmedspace{R}}$, then $(i,P) \in {{\Psi}\negmedspace\mid\negmedspace{R}}$. By the definition of restriction, this implies both $(i,P) \in \Psi$ and $i \in R$. Since $(i,P) \in \Phi$ implies $(i,P) \in \Psi$, we conclude $\Phi \subseteq \Psi$.
\[L:SUBSET\_RESTRICTION\] Let $\Psi$ and $\Phi$ be ensembles. If $\Phi \subseteq \Psi$ and $R = {{\operatorname{dom}{\Phi}}}$, then $\Phi = {{\Psi}\negmedspace\mid\negmedspace{R}}$.
Consider $(i,P) \in {{\Psi}\negmedspace\mid\negmedspace{R}}$. It then follows from the definition of restriction that $(i,P) \in \Psi$ and $i \in R$. But $R = {{\operatorname{dom}{\Phi}}}$, so $i \in {{\operatorname{dom}{\Phi}}}$. This implies there exists $(i,Q) \in \Phi$. Since $\Phi \subseteq \Psi$, then $(i,Q) \in \Psi$. Since $\Psi$ is a mapping, then $(i,P) \in \Psi$ and $(i,Q) \in \Psi$ implies $P = Q$. From $P = Q$ and $(i,Q) \in \Phi$, we infer that $(i,P) \in \Phi$. Thus $(i,P) \in {{\Psi}\negmedspace\mid\negmedspace{R}}$ implies $(i,P) \in \Phi$, so ${{\Psi}\negmedspace\mid\negmedspace{R}} \subseteq \Phi$.
Next suppose $(i,P) \in \Phi$. From this it follows that $i \in R = {{\operatorname{dom}{\Phi}}}$. From the premises $(i,P) \in \Phi$ and $\Phi \subseteq \Psi$ we conclude $(i,P) \in \Psi$. Together $(i,P) \in \Psi$ and $i \in R$ imply that $(i,P) \in {{\Psi}\negmedspace\mid\negmedspace{R}}$. Thus $(i,P) \in \Phi$ implies $(i,P) \in {{\Psi}\negmedspace\mid\negmedspace{R}}$, so $\Phi \subseteq {{\Psi}\negmedspace\mid\negmedspac
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explain how to compute $X\mapsto \sigma(m)^t\cdot h \cdot X+\sigma(X)^t\cdot h \cdot m$ explicitly. Recall that for a $\kappa$-algebra $R$, we denote an element $m$ of $\underline{M}(R)$ by $(m_{i,j}, s_i\cdots w_i)$ with a formal matrix interpretation $m=(\pi^{max\{0, j-i\}}m_{i,j}) \mathrm{~together~with~}z_i^{\ast}, m_{i,i}^{\ast}, m_{i,i}^{\ast\ast}$ (cf. Section \[m\]) and we denote an element $f$ of $\underline{H}(R)$ by $(f_{i,j}, a_i\cdots f_i)$ with a formal matrix interpretation $f=(\pi^{max\{i,j\}}f_{i,j}) \mathrm{~together~with~}f_{i,i}^{\ast}$ (cf. Section \[h\]). Similarly, we can also denote an element $X$ of $T_1(\bar{\kappa})$ by $(m_{i,j}', s_i'\cdots w_i')$ with a formal matrix interpretation $X=(\pi^{max\{0, j-i\}}m_{i,j}') \mathrm{~together~with~}(z_i')^{\ast}, (m_{i,i}')^{\ast}, (m_{i,i}')^{\ast\ast}$ and an element $Z$ of $T_2(\bar{\kappa})$ by $(f_{i,j}', a_i'\cdots f_i')$ with a formal matrix interpretation $Z=(\pi^{max\{i,j\}}f_{i,j}')\mathrm{~together~with~}(f_{i,i}')^{\ast}$. Then we formally compute $X \mapsto \sigma(m^t)\cdot h\cdot X + \sigma(X^t)\cdot h\cdot m$ and consider the reduction of the formal matrix $ \sigma(m^t)\cdot h\cdot X + \sigma(X^t)\cdot h\cdot m$ in a manner similar to that of the reduction explained in Remark \[r35\]. We denote this reduction by $(f_{i,j}'', a_i''\cdots f_i'')$ with a formal matrix interpretation $(\pi^{max\{i,j\}}f_{i,j}'')\mathrm{~together~with~}(f_{i,i}'')^{\ast}$. This $(f_{i,j}'', a_i''\cdots f_i'')$ may and shall be identifed with an element of $T_2(\bar{\kappa})$ in the manner just described. Then $\rho_{\ast, m}(X)$ is the element $Z=(f_{i,j}'', a_i''\cdots f_i'')$ of $T_2(\bar{\kappa})$.\
To prove the surjectivity of $\rho_{\ast, m}:T_1(\bar{\kappa}) \rightarrow T_2(\bar{\kappa})$, it suffices to show the following three statements:
- $X \mapsto h\cdot X $ defines a bijection $T_1(\bar{\kappa}) \rightarrow T_3(\bar{\kappa})$;
- for any $m \in \underline{M}^{\ast}(\bar{\kappa})$, $Y \mapsto \sigma({}^t m) \cdot Y$ defines a biject
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